On Formal Power Series Over Rickart Rings

(1)

On Formal Power Series Over Rickart Rings

Feng-Kuo Huang

Department of Mathematics, National Taitung University Taitung 95002, TAIWAN

Dedicate to Professor P.-H. Lee

Abstract

A ring R is called a (right) Rickart ring if the right annihilator of any element in R is generated, as a right ideal, by an idempotent. This definition is equivalent to that every principle right ideal is projective, and thus a (right) Rickart ring is also known as a (right) PP ring.

Armendariz and Jondrup had shown that if R is a reduced or commutative ring, then the polynomial ring (R[x],+,·) is a PP ring if and only if R is a PP ring. However, this result is not true if the polynomial ring is replaced by the formal power series ring (R[[x]],+,·).

Birkenmeier, Kim and Park had introduced (right) principally quasi-Baer rings as a generalized for Rickart rings. A ring R is callled (right) p.q.-Baer if the right annihilator of a principal right ideal is generated by an idempotent. They also shown that: A ring R is right p.q.-Baer if and only if the polynomial ring (R[x],+,·) is right p.q.- Baer. Again, this result is not true if the polynomial ring is replaced by the formal power series ring (R[[x]],+, .).

In this note, we discuss the conditions that guarantee the Rickart or quasi-Rickart conditions be extended to the foraml power series ring (R[[x]],+,·) or the nearring (R₀[[x]],+,◦).

2000 Mathematics Subject Classification: Primary 16S36, 16W60

Key words and phrases: annihilator ideal, semicentral idempotent, formal power series, principally quasi-Baer ring, Rickart ring

(2)

1st. approach

Introduction and the motivation

The study of Rickart rings has its roots in both functional analysis and homological algebra. In [37] Rickart studied C^∗-algebras with the property that every right annihilator of any element is generated by a projection (an idempotent pis called a projection ifp=p^∗ where∗ is an involution on that algebra). This condition is modified by Kaplansky [29] through introducing Baer rings (a ring is called a Baer ring if the right annihilator of every nonempty subset of R is generated, as a right ideal, by an idempotent of R) to abstract various properties of AW^∗-algebras and von Neumann algebras.

See also Berberian [2] for details.

A ring satisfying a generalization of Rickart’s condition (i.e., every right annihilator of any element in R is generated, as a right ideal, by an idempotent) has a homological characterization as a right PP ring, i.e., every principal right ideal is projective. Left PP rings are defined similarly. A ring R is called a Rickart ring [2, p.18] if it is both right and left PP. Note that a Rickart ring is often refered as a PP ring in literatures. Be aware that PP rings are not initiated as a generalization for Baer rings but instead, a natural derivative from the study of torsion theory. In [21] Hattori investigated two fundamental problems in torsion theory: Is the torsion-freeness of a right module equivalent to the vanishing of its torsion part? Is it possible to divide any right module into its torsion part? The answer is affirmative, it is necessary and sufficient that the ring R is a left PP ring. In a supplement to Hattori’s result, Endo [18] shows that a normal ring R (i.e., any idempotent of R is contained in the center of R) is left PP if and only if it is right PP.

It is natural to ask if some of these properties can be extended from a ring R to the polynomial ring (R[x],+,·) or formal power series ring (R[[x]],+,·) and vice versa. Armendariz [1] and Jøndrup [28] obtained the following results:

Theorem A [1]. Let R be a reduced ring. Then (R[x],+,·) is a PP ring if and only if R is a PP ring.

Theorem 1.2 [28]. Assume R is a commutative ring. Then (R[x],+,·) is PP if and only if R[x] is PP.

(3)

p.q.-Baer) ring if the right annihilator of a principal right ideal is generated by an idempotent. Equivalently, R is right p.q.-Baer if R modulo the right annihilator of any principal right ideal is projective. If R is both right and left p.q.-Baer, then R is said to be p.q.-Baer. It is shown by Birkenmeier, Kim and Park that:

Theorem 3.1 [8]. R is a right p.q.-Baer ring if and only if (R[x],+,·) is a right p.q.-Baer ring.

However, replacing R[x] by R[[x]] is still false in above theorem. Counterex- amples had been given in [8, Example 2.3] and [32, Example 4].

Three commonly used operations for polynomials are addition “ + ”, mul- tiplication “·” and substitution “◦”, respectively. Observe that (R[x],+,·) is a ring and (R[x],+,◦) is a left nearring where the substitution indicates substitution of f(x) into g(x), explicitly f(x) ◦g(x) = g(f(x)) for any f(x), g(x) ∈ R[x]. It is natural to investigate the nearring of polynomials R[x], the zero-symmetric nearring of polynomials R₀[x] and the zero-symmetric nearring of formal power series R0[[x]] when the ring R is equipped with certain annihilator conditions. Motivated by these observations, the author and Birkenmeier [4, 5, 6] initiated the study of various annihilator conditions in the class of nearrings. Let N be a left nearring and S a nonempty subset of N. Denote rAnn_N(S) = {a ∈ N | Sa = 0} and

`Ann_N(S) = {a∈N |aS = 0}. If no confusion will arise, the subscript may be omitted. Let a ∈ N be arbitrary, we define the Rickart-type annihilator conditions [4] in the class of nearrings by describing the following classes:

(1) N ∈ R_r1 if rAnn(a) = eN for some idempotente∈N; (2) N ∈ Rr2 if rAnn(a) = rAnn(e) for some idempotent e∈N; (3) N ∈ R_`1 if Ànn(a) = N e for some idempotente∈N; (4) N ∈ R_`2 if Ànn(a) = Ànn(e) for some idempotent e∈N.

The Rr2 condition is actually considered for ring with involution [2, p.28].

If N is a ring with unity then either one of the above four conditions is equivalent to N being right PP or left PP. Conditions (1) and (3) are direct analogue of the ring case but invoking theRr1condition is severe in a nearring because it requires a right ideal rAnn(a) to be equal to a right N-subgroup eN. A reduced regular nearring with unity certainly satisfies this requirement but it is false in general.

The extension of Baer rings to the nearring of polynomialsR[x],R₀[x] and the nearring of formal power series R₀[[x]] had been investigated in [4, 5, 6].

The extensions of a Rickart ring and its generalizations are barely discussed.

The aim of this paper is to fulfil these investigations for the unsolved cases and questions. The formal power series extension of a Baer ring was studied

(4)

in [5, 9]. Thus it is natural to ask: What can be said about various Rickart- type annihilator conditions for power series under addition and substitution?

Unfortunately, the substitution of one formal power series into another may have no meaning in general. We use f(x) to denote the formal power series P∞

i=0f_ixⁱ where f_i are in a ring R with unity and f(x)◦ g(x) indicates substitution of f(x) into g(x). We use R[[x]] to denote the set of formal power series over a ring R. Let f(x) = P∞

i=0f_ixⁱ and (x)g = P∞ j=0g_jx^j. Observe that

f(x)◦g(x) = g₀+

∞

X

i=1

g_if₀ⁱ

! +

∞

X

p=1

∞

X

j=1

g_jC_p^(j)

! x^p, where C_p^(j) = X

u1+···+u_j=p

f_u₁f_u₂· · ·f_u_j for p∈ {1,2,3,· · · }.

A necessary requirement for f(x) ◦ g(x) to be well defined is that g₀ + P∞

i=1g_if₀ⁱ ∈ R. One way to solve this problem is to introduce a topology on the ring R. Of particular interest is when R is the complex number field C. The entire function has a unique power series expansion at 0, the substitution is well defined on the power series expansions of entire functions [14].

Another approach is to consider the formal power series with zero constant terms. Note that assuming f₀ = g₀ = 0 then Cp^(j) = 0 when j > p and so the coefficient of each term in the expression of f(x)◦g(x) will be a finite sum of elements from R. Hence the operation of substitution on these power series is well defined. In the sequel, the collection of all power series with zero constant terms using the operations of addition and substitution is denoted by R₀[[x]] unless specifically indicated otherwise (i.e., R₀[[x]] denotes (R₀[[x]],+,◦)). Observe that the system (R₀[[x]],+,◦) is a 0-symmetric abelian nearring. Hence the theory of nearrings provides a natural framework in which to study power series under the operations of addition and substitution. For expositions using this approach one may refer to [16, 30, ?].

Moreover, there has been recent interest by group theorists in pro-p groups and the Nottingham groups [13, 31]. In fact the group studied in [26, 27] is a normal subgroup of the group of units of (R₀[[x]],+,◦) when R is a com-

(5)

Another reason (or motivation?)

Theorem 1. Let R be an IFP ring with unity. Then the following are equivalent.

(1) R is Rickart;

(2) (R[x],+,◦)∈ R_r2; (3) (R₀[x],+,◦)∈ R_r2; (4) (R[x],+,·) is Rickart.

The following example shows that assuming R an IFP ring is not superfluous.

Example 2. There is a Rickart ring R such that the ring (R[x],+,·) is not Rickart and the nearring (R₀[x],+,◦)6∈ R_r2. LetRbe the 2-by-2 matrix ring over the ring of integersZ. Then Ris a Baer ring (hence a Rickart ring), but (R[x],+,·) is not right (left) Rickart (hence not Rickart) because the right annihilator of (²_{0 0}^x) and the left annihilator (^{2 0}_x₀) are not generated by an idempotent, respectively. This example (due to P.M. Cohn) is presented in [1, 8, 28].

Next, we will show that the nearring (R₀[x],+,◦)6∈ R_r2(hence (R[x],+,◦)6∈

R_r2). Let η(x) =Pm

i=1η_ixⁱ be an idempotent in (R₀[x],+,◦). It is immediate that η₁² =η1 is an idempotent inR. Ifη(x) is a nonzero idempotent, then η₁ 6= 0. Otherwise, if η(x) =Pm

i=2η_ixⁱ then, by comparing coefficients from the equation η(x)◦η(x) =η(x), it implies that η₂ = 0. Inductively, it shows η(x) = 0. Hence η1 6= 0. It is now not difficult to see that the polynomial x² 6∈rAnn(η(x)) since

η(x)◦x² =η(x)·η(x) =η₁x²+





 X

i+j=k 16i,j6m

η_iη_j





x^k.

Let f(x) = (^{0 1}_{0 0})x ∈R₀[x]. Then x² ∈rAnn(f(x)) but x² 6∈rAnn(η(x)) for any idempotent in the nearring (R₀[x],+,◦). Thus (R₀[x],+,◦)6∈ R_r2.

(6)

Nearring of formal power series

Lemma 3. Let R be an IFP ring with unity. If η(x) = P∞

i=1η_ixⁱ ∈ R₀[[x]]

is an idempotent, then η(x) = η₁x with η²₁ =η₁.

Proposition 4. Let R be an IFP ring with unity. If R0[[x]]∈ Rr1∪ Rr2∪ R_`1∪ R_`2, then R is a Rickart ring.

Proposition 5. Let R be a finite reduced ring with unity. If R is a Rickart ring, then R0[[x]]∈ Rr1∪ Rr2∪ R`1∪ R`2.

In the above Propositions, the existence of unity is assumed but ,in fact, it is not quite necessary. The following example shows that assuming R a finite reduced ring is not superfluous.

Example 6. There exists a commutative reduced ring R such that R ∈ R_r2∪R_`2 but (R₀[[x]],+,◦)6∈ R_r1∪R_r2∪R_`1∪R_`2. Indeed, letR =L

Z2, a direct sum of infinitely many copies of the fieldZ². Note that each element in Ris an idempotent, soR∈ R_r2∪R_`2. Lete₁ = (1,0,0,· · ·),e₂ = (0,1,0,· · ·),

· · ·,e_j = (0,· · · ,0,1,0,· · ·),· · · ∈R, wheree_j denotes the element inRwith j^th-component equals to 1 and 0 elsewhere for all j ∈ N. Observe that all the monomials ax are idempotents in R₀[[x]] and vice versa. Since R₀[[x]]

is reduced [5, Proposition 3.1], and thus all the idempotents are central. It is not difficult to see that `Ann(ax) = rAnn(ax) 6= {0}. For instance, if a = (1,1,0,· · ·) and b = (0,0,1,0,· · ·), then bx ∈ `Ann(ax) = rAnn(ax).

Explicitely, if j is the largest number such that the j^th-component of a∈ R is nonzero, then e_j+1x∈`Ann(ax) = rAnn(ax).

Let f(x) = P∞

i=1e_ixⁱ ∈ R₀[[x]]. If g(x) = P∞

j=1g_jx^j ∈ rAnn(f(x)), then g_je_i = 0 for all i, j ∈ N by [5, Lemm 3.3] and thus g_j = 0 for all j ∈ N. Therefore rAnn(f(x)) = 0. Similarly, `Ann(f(x)) = 0 and thus (R₀[[x]],+,◦)6∈ R_r2 ∪ R_`2.

On the other hand, let h(x) =P∞

i=1e_2ix²ⁱ. Then e2i−1x∈ rAnn(h(x))∩

`Ann(h(x)) for alli∈N. SinceaRis finite andR₀[[x]]◦(ax) = (ax)◦R₀[[x]] = (aR)₀[[x]], it follows that rAnn(h(x)) 6= (ax) ◦R₀[[x]] and `Ann(h(x)) 6=

R₀[[x]]◦(ax) for all a∈R. Thus (R₀[[x]],+,◦)6∈ R_r1∪ R_`1.

(7)

Theorem 7. Let R be a finite reduced ring with unity. Then the following are equivalent.

(1) R is Rickart;

(2) (R[x],+,◦)∈ R_r2; (3) (R₀[x],+,◦)∈ R_r2; (4) (R₀[[x]],+,◦)∈ R_r2; (5) (R[x],+,·) is Rickart.

The following example shows that assumingRa finite ring in above result is not superfluous.

Example 8. There exists a reduced ringT with unity such thatT is a Rickart ring but (T₀[[x]],+,◦)6∈ R_r2. LetR=Q

Z2 ={a: N→Z2}be the ring of direct product of the fieldZ2. LetT ={a∈R|a(i) is eventually constant for i∈ N}. ThenT is a proper subring ofR. Observe thatT is a reduced ring with unity 1. Since each element in T is an idempotent, it is a Rickart ring. In fact, rAnn_T(a) = (1−a)T. Let e_i ∈T such that

e_i(j) =

1 ifi=j, 0 ifi6=j;

and f(x) = P∞

i=1e_2ix²ⁱ. Observe that e_se_t = 0 if s 6= t ∈ N. Then g(x) = P∞

i=1e2i−1x²ⁱ⁻¹ ∈rAnn(f(x)) =`Ann(f(x)) by [5, Lemma 3.3]. By Lemma 4.1, idempotents in T₀[[x]] are ax for all a∈T and

rAnn(ax) ={

∞

X

i=1

hixⁱ |hi ∈(1−a)T}.

It is immediate that g(x)6∈rAnn(ax) for alla ∈T. Thus T₀[[x]]6∈ R_r2.

(8)

Formal power series ring

It is known that even reduced Rickart ring is not stable when extending to the formal power series ring (R[[x]],+,·). In fact, counterexamples had been provided to show that (R[[x]],+,·) is not Rickart when R is a commutative von Neumann regular ring [8, Example 2.3] and [32, Example 4].

However, Theorem 4.5 did motivate the following question: Is it true that the formal power series ring (R[[x]],+,·) a Rickart ring when R is a finite reduced Rickart ring? The answer is affirmative as presented in the following discussions. Be aware that in the following we are studying the formal power series ring (R[[x]],+,·) instead of the the formal power series nearring (R₀[[x]],+,◦).

Lemma 9. Let R be an IFP ring with unity. If η(x)is an idempotent in the formal power series ring (R[[x]],+,·), thenη(x) =e is a constant polynomial where e is an idempotent in the ring R.

Lemma 10. LetRbe a reduced ring andf(x) =P∞

i=0f_ixⁱ,g(x) = P∞

j=0g_jx^j ∈ (R[[x]],+,·). Thenf(x)·g(x) = 0if and only iffigj = 0 for alli, j ∈N∪{0}.

Theorem 11. Let R be a finite reduced ring with unity. Then R is Rickart if and only if the formal power series ring (R[[x]],+,·) is Rickart.

The following example shows that assumingR finite is not superfluous.

Example 12. There is a commutative reduced ringRwith unity such thatR is a Rickart ring but the formal power series ring (R[[x]],+,·) is not Rickart.

Let F be a field and let R={(a_n)^∞_n=1 ∈

∞

Y

n=1

F_n |a_n is eventually constant}, a subring of Q∞

n=1F_n where F_n=F for all n∈N. This ring R is a commutative von Neumann regular ring (hence a reduced Rickart ring) but the ring (R[[x]],+,·) is not Rickart [8, Example 2.3]. See also [32, Example 4] for a commutative Rickart ring R but (R[[x]],+,·) is not Rickart.

(9)

2nd. approach

Throughout this note, R denotes a ring with unity. Recall that R is called a (quasi-)Baer ring if the right annihilator of every (right ideal) nonempty subset of R is generated, as a right ideal, by an idempotent ofR. Baer rings are introduced by Kaplansky [29] to abstract various properties ofAW^∗-algebras and von Neumann algebras. Quasi-Baer rings, introduced by Clark [15], are used to characterize when a finite dimensional algebra over an algebraically closed field is isomorphic to a twisted matrix units semigroup algebra. The definition of a (quasi-) Baer ring is left-right symmetric [15, 29].

In [10], Birkenmeier, Kim and Park initiated the study of right principally quasi-Baer rings. A ring R is called right principally quasi-Baer (or simply right p.q.-Baer) if the right annihilator of a principal right ideal is generated, as a right ideal, by an idempotent. Equivalently, R is right p.q.-Baer if R modulo the right annihilator of any principal right ideal is projective. If R is both right and left p.q.-Baer, then it is called p.q.-Baer. The class of p.q.- Baer rings include all biregular rings, all quasi-Baer rings and all abelian PP rings. See [10] for more details.

Ore extensions or polynomial extensions of (quasi-)Baer rings and their generalizations are extensively studied recently ([4] to [11] and [22] to [25]).

It is proved in [9, Theorem 1.8] that a ring R is quasi-Baer if and only if R[[X]] is quasi-Baer, whereX is an arbitrary nonempty set of not necessarily commuting indeterminates. In [8, Theorem 2.1], it is shown that R is right p.q.-Baer if and only ifR[x] is right p.q.-Baer. But it is not equivalent to that R[[x]] is right p.q.-Baer. In fact, there exists a commutative von Neumann regular ring R (hence p.q.-Baer) such that the ringR[[x]] is not p.q.-Baer [8, Example 2.6]. In [33, Theorem 3], a necessary and sufficient condition for semiprime ring under which the ring R[[x]] is right p.q.-Baer are given. It is shown that R[[x]] is right p.q.-Baer if and only if R is right p.q.-Baer and any countable family of idempotents inR has a generalized join when all left semicentral idempotents are central. Indeed, for a right p.q.-Baer ring, asking the set of left semicentral idempotents S_`(R) equals to the set of central idempotents B(R) is equivalent to assume R is semiprime [10, Proposition 1.17]. In this note, the condition requiring all left semicentral idempotents being central is shown to be redundant. We show that: The ring R[[x]] is right p.q.-Baer if and only if R is p.q.-Baer and every countable subset of right semicentral idempotents has a generalized countable join. This theorem properly generalizes Fraser and Nicholson’s result in the class of reduced PP rings [19, Theorem 3] and Liu’s result in the class of semiprime p.q.-Baer rings [33, Theorem 3]. For simplicity of notations, denote N ={0,1,2,· · · } be the set of natural numbers.

(10)

Annihilators and left semicentral idempotents Lemma 13. Let f(x) = P∞

i=0f_ixⁱ, g(x) = P∞

j=0g_jx^j ∈ R[[x]]. Then the following are equivalent.

(1) f(x)R[[x]]g(x) = 0;

(2) f(x)Rg(x) = 0;

(3) P

i+j=kf_iag_j = 0 for all k∈N, a∈R.

Recall that an idempotent e ∈ R is called left (resp. right) semicentral [3] if re = ere (resp. er = ere) for all r ∈ R. Equivalently, e = e² ∈ R is left (resp. right) semicentral if eR(resp.Re) is an ideal ofR. Since the right annihilator of a right ideal is an ideal, we see that the right annihilator of a right ideal is generated by a left semicentral idempotents in a right p.q.-Baer ring. The set of left (resp. right) semicentral idempotents of R is denoted S`(R) (resp. Sr(R)). The following result is used frequently later in this note.

Lemma 14. [10, Lemma 1.1] Lete be an idempotent in a ringR with unity.

Then the following conditions are equivalent.

(1) e∈ S`(R);

(2) 1−e∈ S_r(R) ; (3) (1−e)Re= 0;

(4) eR is an ideal of R;

(5) R(1−e) is an ideal ofR.

To prove the main result, we first characterize the left semicentral idempotents in R[[x]].

Proposition 15. Let ε(x) = P∞

i=0ε_ixⁱ ∈ R[[x]]. Then ε(x) ∈ S_`(R[[x]]) if and only if

(1) ε₀ ∈ S_`(R);

(2) ε₀rε_i =rε_i and ε_irε₀ = 0 for all r∈R, i= 1,2,· · · ; (3) P

i+j=k

i,j>1 ε_irε_j = 0 for all r∈R and k >2.

Corollary 16. [8, Proposition 2.4(iv)] LetR be a ring with unity and ε(x) = P∞

i=0ε_ixⁱ ∈ S_`(R[[x]]). Then ε(x)R[[x]] =ε₀R[[x]].

(11)

Generalized countable join

Let R be a ring with unity and E = {e₀, e₁, e₂,· · · } a countable subset of S_r(R). We say E has a generalized countable join e if, given a∈R, there exists e∈ S_r(R) such that

(1) e_ie=e_i for all i∈N;

(2) ife_ia=e_i for all i∈N, then ea=e.

Note that if there exists an element e ∈ R satisfies conditions (1) and (2) above, then e ∈ S_r(R). Indeed, the condition (1): e_ie = e_i for all i ∈ N implies ee = e by (2) and so e is an idempotent. Further, let a ∈ R be arbitrary. Then the element d = e −ea +eae is an idempotent in R and e_id=e_i for alli∈N. Thused=eby (2). Note thated=e(e−ea+eae) =d.

Consequently, e=d=e−ea+eae or ea=eae. Thus e∈ S_r(R).

Note that a generalized countable join e, if it exists, is indeed a join if S_r(R) is a lattice. Recall that when R is an abelian ring (i.e., every idempotent is central), then the set B(R) = S_r(R) of all idempotents in R is a Boolean algebra where e 6 d means ed = e. Let e be a join of E ={e₀, e₁, e₂,· · · }inB(R) whereRis a reduced PP ring. That isesatisfies (1) e_ie = e_i for all i ∈ N; (2⁰) if e_id = e_i for all i ∈ N and any d ∈ B(R), then ed = e. Given an arbitrary a ∈ R, then 1−a = pu for some central idempotentp∈R and someu∈R such thatrAnn_R(u) = 0 =`Ann_R(u) [19, Proposition 2]. Observe that if e_ia=e_i for alli∈N, then e_i(1−a) =e_ipu= 0. It follows that e_ip= 0 for all i∈ N since `Ann_R(u) = 0. Thus ep= 0 or e(1−a) = epu = 0. Therefore ea =e and e is a generalized countable join of E. In other words, a generalized countable join is a join and vice versa in the class of reduced PP rings.

Be aware that (S_r(R),6) is not partially ordered by defining d6ewhen de = d in an arbitrary ring R. This relation is reflexive, transitive but not antisymmetric. However, let a, b ∈ S_r(R) and define a ∼ b if a = ab and b = ba. Then ∼ is an equivalence relation on S_r(R) and (S_r(R)/∼,6) is a partially ordered set. In the case when (S_r(R)/ ∼,6) is a complete lattice, then a generalized countable join exists for any subset ofS_r(R). In particular whenRis a Boolean ring or a reduced PP ring, then the generalized countable join is indeed a join in R.

In [33, Definition 2], Liu defined the notion of generalized join for a countable set of idempotents. Explicitely, let {e₀, e₁,· · · }be a countable family of idempotents of R. The set {e₀, e₁,· · · }is said to have a generalized join e if there exists e=e² such that

(i) e_iR(1−e) = 0;

(ii) if d is an idempotent and e_iR(1−d) = 0 then eR(1−d) = 0.

Observe that

e_ir(1−e) = e_ire_i(1−e) = e_ir(e_i−e_ie),

(12)

when e_i ∈ S_r(R). Thus e_i = e_ie if and only if e_ir(1−e) = 0 for all r ∈ R when e_i ∈ S_r(R) for alli∈N. Now, letE ={e₀, e₁, e₂,· · · } ⊆ S_r(R) andea generalized countable join ofE. To showe is a generalized join (in the sense of Liu), it remains to show condition (ii) holds. Let f be an idempotent in R such thate_iR(1−f) = 0. Then, in particular,e_i(1−f) = 0 for all i∈N. Thus e(1−f) = 0 by hypothesis. It follows that er(1−f) = ere(1−f) = 0 and thus eR(1−f) = 0. Therefore, e is a generalized join of E. Thus, in the content of right semicentral idempotents, a generalized countable join is a generalized join in the sense of Liu.

Conversely, lete∈ S_r(R) be a generalized join (in the sense of Liu) of the setE ={e₀, e₁, e₂,· · · } ⊆ S_r(R). Observe that condition (ii) is equivalent to

(ii⁰) ifd is an idempotent and e_id =e_i then ed =e.

Let a ∈ R be arbitrary such that e_ia = e_i for all i ∈ N. Then condition (ii’) and a similar argument used in the case of reduced PP rings implies that ea = e. Thus e is a generalized countable join. Therefore, in the content of right semicentral idempotents, Liu’s generalized join is equivalent to generalized countable join.

(13)

Main Result

Theorem 17. Let R be a ring with unity. Then R[[x]] is right p.q.-Baer if and only if R is right p.q.-Baer and every countable subset of S_r(R) has a generalized countable join.

Since Liu’s generalized join is equivalent to generalized countable join in the set of right semicentral idempotents S_r(R). The following result is immediated from Theorem 5.

Corollary 18. [33, Theorem 3] Let R be a ring such that S_`(R) ⊆ B(R).

Then R[[x]] is right p.q.-Baer if and only if R is right p.q.-Baer and any countable family of idempotents in R has a generalized join.

Corollary 19. [19, Theorem 3] IfRis a ring then R[[x]]is a reduced PP ring if and only ifR is a reduced PP ring and any countable family of idempotents in R has a join in B(R).

(14)

3rd. Approach

Introduction

Recall from [29] that R is a Baer ring if R has a unity and the right annihilator of every nonempty subset of R is generated, as a right ideal, by an idempotent. The study of Baer rings has its roots in functional analysis ([2] and [29]). For example, every von Neumann algebra (e.g., the algebra of all bounded linear operators on a Hilbert space) is a Baer ring. Kaplansky shows in [29] that the definition of a Baer ring is left-right symmetric.

In [4], various annihilator conditions on polynomials under addition and substitution are investigated. The formal power series extension of a Baer ring was studied in [9]. Thus it is natural to ask: What can be said about various Baer-type annihilator conditions for power series under addition and substitution? Unfortunately, the substitution of one formal power series into another may have no meaning in general. We use (x)f to denote the formal power series P∞

i=0f_ixⁱ where f_i are in a ring R with unity and (x)f ◦(x)g indicates substitution of (x)f into (x)g. Composition of functions is per- formed in a similar manner. We use R[[x]] to denote the set of formal power series over a ring R. Let (x)f =P∞

i=0f_ixⁱ and (x)g = P∞

j=0g_jx^j. Through a lengthy calculation, we have

(x)f ◦(x)g = ((x)f)g = g₀+

∞

X

i=1

g_if₀ⁱ

! +

∞

X

p=1

∞

X

j=1

g_jc^(j)_p

! x^p, wherec^(j)_p = X

u1+···+uj=p

f_u₁f_u₂· · ·f_u_j for p∈ {1,2,3,· · · }.

A necessary requirement for (x)f ◦ (x)g to be well defined is that g₀ + P∞

i=1g_if₀ⁱ ∈ R. One way to solve this problem is to introduce a topology on the ring R. Of particular interest is when R is the complex number field C. Recall a function f :D → C is called analytic on an open set D ⊆ C if f has a power series expansion at every point in D. It is called entire if f is analytic on C. Since the composition of two entire functions is entire [14, Proposition 5.1, p.22] and each entire function has a unique power series expansion at 0, the substitution is well defined on the power series expansions

(15)

of substitution on these power series is well defined. In the sequel, the collection of all power series with positive orders using the operations of addition and substitution is denoted by R₀[[x]] unless specifically indicated otherwise (i.e., R₀[[x]] denotes (R₀[[x]],+,◦)). Observe that the system (R₀[[x]],+,◦) is a 0-symmetric left nearring. Hence the theory of nearrings provides the natural framework in which to study power series under the operations of addition and substitution. For expositions using this approach one may refer to Clay [16], Kautschitsch and Mlitz [30]. Moreover, there has been recent interest by group theorists in pro-p groups and the Nottingham groups [?,?].

In fact the group studied in [?,?] is a normal subgroup of the group of units of (R₀[[x]],+,◦) when R is a commutative ring, and the Nottingham group is a normal subgroup of the group of units of (R₀[[x]],+,◦) whenR is a finite field of characteristic p > 2. Note that power series without constant terms often arise as a unique formal solution to the differential equation ^dy_dx = f where f is an analytic function over C² [14, pp.210–212].

Throughout this paper all rings are associative and all nearrings are left nearrings. We useRandN to denote a ring and nearring, respectively. For a nonemptyS ⊆N,r_N(S) ={a ∈N|Sa= 0}and`_N(S) = {a∈N|aS = 0}.

If the context is clear, the subscript may be omitted. Baer-type annihilator conditions in the class of nearrings have been defined in [?] as the follows:

(1) N ∈ B_r1 if the right annihilator r(S) =eN for some idempotent e∈N; (2) N ∈ B_r2 if the right annihilator r(S) =r(e) for some idempotent e∈N; (3) N ∈ B_`1 if the left annihilator `(S) = N e for some idempotente∈N; (4) N ∈ B_`2 if the left annihilator `(S) = `(e) for some idempotent e∈N.

When S is a singleton, the Rickart-type annihilator conditions on nearrings are also defined and denoted similarly except B is replaced by R. In [2, p.28], the R_r2 condition is considered for rings with involution. If N is a ring with unity then N ∈ B_r1∪ B_r2∪ B_`1∪ B_`2 is equivalent to N being a Baer ring.

(16)

Analytical approach to formal power series

The ring of analytic functions (E(D),+,·) over a connected open set D is an integral domain [14, Corollary 1, p.40]. However this does not apply to the nearring E(C) as shown in the following proposition.

Proposition 20. Let f, g ∈ E(C) be two entire functions where g 6= 0. If f ◦g = 0, then f is a constant.

Corollary 21. The idempotents inE(C) are either constants or the identity function µ.

Theorem 22. Let E(C) be the nearring of entire functions. Then E(C) ∈ Rr2 but not in Br2.

A nearring N is called integral if N has no nontrivial zero divisors. We have seen the nearringE(C) is not integral. However, its subnearringE₀(C) = {f ∈ E(C)|(0)f = 0}is indeed integral, as a consequence of Proposition 2.1.

Thus E₀(C) satisfies all the Baer-type annihilator conditions as expected by [?, Proposition 1.8]. We write these observations in the following result.

Proposition 23. The 0-symmetric nearring E₀(C) is integral with unity µ, and E₀(C)∈ B_r1∩ B_r2∩ B_`1∩ B_`2.

(17)

Algebraic approach to formal power series

We now study Baer-type annihilator conditions on formal power series via the algebraic approach. Throughout this section, R₀[[x]] will denote the nearring of formal power series (R₀[[x]],+,◦) with positive orders. If S ⊆ R₀[[x]] thenr(S) = {f ∈R₀[[x]]|S◦f = 0}and`(S) ={f ∈R₀[[x]]|f◦S= 0}.

Proposition 24. Let R be a ring. Then R is reduced if and only if R₀[[x]]

is reduced.

Lemma 25. Let R be a ring. If (x)η =P∞

i=1η_ixⁱ ∈R₀[[x]]is an idempotent, then η₁² =η₁. If R is reduced, then (x)η=η₁x.

Lemma 26. Let R be a reduced ring and (x)f,(x)g ∈ R₀[[x]] with (x)f = P∞

i=1f_ixⁱ and(x)g =P∞

j=1g_jx^j. Then (x)f◦(x)g = 0 if and only if g_jf_i = 0 for all i, j ∈ {1,2,3,· · · }.

If (x)f =P∞

i=1f_ixⁱ ∈R₀[[x]], let S_f^∗ :={f_i|i∈N}.

Proposition 27. Let R be a reduced ring. Then (1) R ∈ B_r1 if and only if R₀[[x]]∈ B_`1; (2) R ∈ B_r2 if and only if R₀[[x]]∈ B_`2. Theorem 28. Let R be a reduced ring.

(1) If R is Baer, then R₀[[x]]∈ B_r1∩ B_r2∩ B_`1∩ B_`2. (2) If R₀[[x]]∈ B_r1∪ B_r2∪ B_`1∪ B_`2 , then R is Baer.

Corollary 29. Let R be a reduced ring. The following are equivalent:

(1) R is Baer ;

(2) (R[[x]],+,·) is Baer ;

(3) (R₀[[x]],+,◦)∈ B_r1∪ B_r2∪ B_`1∪ B_`2; (4) (R₀[[x]],+,◦)∈ B_r1∩ B_r2∩ B_`1∩ B_`2.

Proposition 30. Assume R is a reduced ring. Let S be the subnearring of R₀[[x]]generated by the set {ex|e=e² ∈R}and T a subnearring of R₀[[x]].

If R₀[[x]]∈ B_νi, where ν∈ {r, `} and i∈ {1,2}, and S ⊆T, then T ∈ B_νi. Example 31. Using Proposition 3.4, the following nearrings satisfy all the Baer-type annihilator conditions discussed in this paper when Ris a reduced Baer ring. (i) {ax|a ∈ R}; (ii) {(x)f = P∞

i=1a2i−1x²ⁱ⁻¹ ∈ R₀[[x]]|a2i−1 ∈ R for all i∈N}; (iii)E₀[[x]], whereE is a subring containing all idempotents of R.

Corollary 32. The nearring of 0–preserving entire functions E₀(C)∈ B_r1∩ B_r2∩ B_`1∩ B_`2.

(18)

Since there is both a ring and nearring structure onR₀[[x]], it is natural to ask: What are the connections between the ring structure (R₀[[x]],+,·) and the nearring structure (R₀[[x]],+,◦)? Our remaining results address this question.

Let N be a nearring and 0 ⊆ X ⊆ Y ⊆ N. We say X is 2-essential in Y, denoted by X ≤^ess₂ Y, if for each nonzero N-subgroup I, I ⊆ Y implies X ∩ I 6= 0. We use X C N to denote that X is an ideal of N. From [1, Lemma 1] and Lemma 3.3, if (x)g right (left) annihilates (x)f in (R₀[[x]],+,·) then (x)g right (left) annihilates (x)f in (R₀[[x]],+,◦) when R is reduced. Moreover, the following result shows that if R is a reduced Baer ring then every nearring ideal of R₀[[x]] is 2-essential in a nearring direct summand which is also a ring direct summand. Note that if e is a central idempotent in a ring R, then ex ◦ R₀[[x]] C (R₀[[x]],+,◦) and ex ◦R₀[[x]] = e ·R₀[[x]] C (R₀[[x]],+,·). Moreover, all idempotents in a reduced ring or nearring with unity are central.

Theorem 33. Let R be a reduced Baer ring and 0 6= I C (R₀[[x]],+,◦).

Then there exists e =e² ∈ R such that I ≤^ess₂ ex◦R₀[[x]] C (R₀[[x]],+,◦), and I∩B 6= 0 for every 0 6= B C (R₀[[x]],+,·) such that R ·B ⊆ B and B ⊆ e·R₀[[x]]. Moreover, if I C(R₀[[x]],+,·), then I is essential as a ring right ideal in the ring direct summand e·R₀[[x]] of the ring (R₀[[x]],+,·).

Corollary 34. Let R be a reduced Baer ring.

(1) If % is a radical map, then R₀[[x]] = A⊕S (nearring direct sum), where %(R0[[x]])≤^ess₂ A and S is %-semisimple.

(2) If M is a maximal ideal of R₀[[x]], then M ≤^ess₂ R₀[[x]].

Recall (R,+,·,◦) is called acomposition ring [16, 30] if (R,+,·) is a ring and (R,+,◦) is a left nearring satisfying a◦(b·c) = (a◦b)·(a◦c) for all a, b, c∈R. Thus (R₀[[x]],+,·,◦) is a composition ring if R is a commutative ring. With the following definitions, Theorem 3.10 can be applied to the case when R₀[[x]] is a composition ring. A subset I of R is called a full ideal of R if I is both a ring ideal and a nearring ideal of R. We call a nonzero full idealI of a composition ring (R,+,·,◦)right 2-essential in a subcomposition ring T of R, if I has nonzero intersection with every nonzero right ideal of (R,+,·) which is contained in T and I has nonzero intersection with every

(19)

References

[1] E.P. Armendariz, A note on extensions of Baer and p.p.-rings,J. Austral.

Math. Soc. 18 (1974), 470–473.

[2] S.K. Berberian, Baer ∗-rings, Springer Verlag, Berlin, 1972.

[3] G.F. Birkenmeier, Idempotents and completely semiprime ideals, Comm. Algebra 11 (1983), 567–580.

[4] G.F. Birkenmeier, F.-K. Huang, Annihilator conditions on polynomials, Comm. Algebra 29 (2001), 2097–2112.

[5] G.F. Birkenmeier, F.-K. Huang, Annihilator conditions on formal power series, Algebra Colloq. 9 (2002), 29–37.

[6] G.F. Birkenmeier, F.-K. Huang, Annihilator conditions on polynomials II, Monatsh. Math. 141 (2004), 265–276.

[7] Y. Cheng and F.-K. Huang, A note on extensions of principally quasi- Baer rings, Taiwanese J. Math. 12 (2008), 1721–1731.

[8] G.F. Birkenmeier, J.Y. Kim, J.K. Park, On polynomial extensions of principally quasi-Baer rings, Kyungpook Math. J.40 (2000), 247–253.

[9] G.F. Birkenmeier, J.Y. Kim, J.K. Park, Polynomial extensions of Baer and quasi-Baer rings, J. Pure Appl. Algebra 159 (2001), 25–42.

[10] G.F. Birkenmeier, J.Y. Kim, J.K. Park, Principally quasi-Baer rings, Comm. Algebra 29 (2001), 639–660.

[11] G.F. Birkenmeier, J.K. Park, Triangular matrix representations of ring extensions, J. Algebra 265 (2003), 457–477.

[12] G. F. Birkenmeier, J. K. Park, and S. T. Rizvi, Quasi-Baer Hulls, preprint.

[13] R. Camina, Subgroups of the Nottingham group, J. Algebra 196(1997) 101–113.

[14] H. Cartan, Theory of Analytic Functions, Addison-Wesley, Reading, 1963.

[15] W.E. Clark, Twisted matrix units semigroup algebras, Duke Math. J.

34 (1967), 417–424.

(20)

[16] J. R. Clay, Nearrings: Geneses and Applications, Oxford Univ. Press Inc., Oxford, 1992.

[17] N. Divinsky, Rings and Radicals, University of Toronto Press, Toronto, Ont., 1965.

[18] S. Endo, Note on p.p. rings (A supplement to Hattori’s paper), Nagoya Math. J. 17 (1960) 167–170.

[19] J.A. Fraser, W.K. Nicholson, Reduced PP-rings, Math. Japonica 34 (1989), 715–725.

[20] K.R. Goodearl, Von Neumann Regular Rings, Krieger, Malabar, 1991.

[21] A. Hattori, A foundation of torsion theory for modules over general rings, Nagoya Math. J. 17 (1960) 147–158.

[22] Y. Hirano, On annihilator ideals of a polynomial ring over a commutative ring, J. Pure Appl. Algebra 168 (2002), 45–52.

[23] C.Y. Hong, N.K. Kim, T.K. Kwak, Ore extensions of Baer and p.p.- rings, J. Pure Appl. Algebra 151 (2000), 215–226.

[24] F.-K. Huang, On polynomials over Rickart rings and their generalizations I, Monatsh. Math.151 (2007), 45–65.

[25] C. Huh, H.K. Kim, Y. Lee, p.p. rings and generalized p.p. rings,J. Pure Appl. Algebra 167 (2002), 37–52.

[26] S.A. Jennings, Substitution groups of formal power series, Canad. J.

Math. 6 (1954) 325–340.

[27] D.L. Johnson, The group of formal power series under substition, J.

Austral. Math. Soc. 45 (1988) 296–302.

[28] S. Jøndrup, p.p. rings and finitely generated flat ideals, Proc. Amer.

Math. Soc. 28 (1971) 431–435.

(21)

[32] Y. Lee, C. Huh, Counterexamples on p.p.-rings, Kyungpook math. J.38 (1998), 421–427.

[33] Z. Liu, A note on principally quasi-Baer rings, Comm. Algebra 30 (2002), 3885–3990.

[34] A.C. Mewborn, Regular rings and Baer rings, Math. Z. 121 (1971), 211–219.

[35] A. Moussavi, H. Haj Seyyed Javadi, E. Hashemi, Generalized quasi-Baer ring, Comm. Algebra 33 (2005), 2115–2129.

[36] G. Pilz, Near-rings, North Holland, Amsterdam, 2nd revised ed., 1983.

[37] C.E. Rickart, Banach algebras with an adjoint operations,Ann. of Math.

47 (1946), 528–550.

[38] L.W. Small, Semihereditary rings, Bull. Amer. Math. Soc. 73 (1967), 656–658.