An idempotenteofRisprimitiveif there does not exist an idempotent f ∈R such that ef =f and f e6=e

THE IDEMPOTENT QUIVER OF A NEARRING

GARY PETERSON

JAMES MADISON UNIVERSITY, HARRISONBURG, VIRGINIA USA

R denotes a 0-symmetric left nearring with 1.

Assume R satisfies dcc on right R-subgroups and J₂(R) is nilpotent.

An idempotenteofRisprimitiveif there does not exist an idempotent f ∈R such that ef =f and f e6=e.

A right R-subgroup M of R is self-monogenic if mM =M for some m∈M.

Theorem 1. Lete be an idempotent of R. TFAE:

1. e is primitive.

2. eR is a minimal self-monogenic rightR-subgroup of R.

3. eR is a minimal nonnilpotent rightR-subgroup of R.

A set of idempotents e₁, . . . , e_n of R is principal if for any r ∈ R = R/J₂(R),

r =e₁r+· · ·+e_nr.

A PPO-set is a principal set of primitive orthogonal idempotents.

Theorem 2. Suppose that I is a nilpotent ideal of R and ε₁, . . . , ε_n is a set of primitive orthogonal idempotents of R = R/I. Then there exists a set of primitive orthogonal idempotents

e₁, . . . , e_n of R such that e_i =ε_i. Theorem 3. PPO-sets exist.

Outline of Proof. Let

R =R/J₂(R) =A₁⊕ · · · ⊕A_n

be Wedderburn decomposition of R into minimal right ideals. Write 1 = ε1+· · ·+εn,

ε_i ∈A_i. Now lift ε₁, . . . , ε_n to set of primitive orthogonal idempotents

e₁, . . . , e_n of R.

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Theorem 4. A set of primitive orthogonal idempotents e1, . . . , en is a PPO-set ⇔ Ann_R(e₁, . . . , e_n) is nilpotent.

Theorem 5. A nonnilpotent right R-subgroup of R contains a primi- tive idempotent.

Def. Two primitive idempotents e and f are linked if there exist primitive idempotents

e=e₁, e₂, . . . , e_n =f

such that e_iR and e_i+1R have isomorphic R-factors for eachi.

Theorem 6. Let e1 and e2 be primitive idempotents of R. e1R and e₂R have isomorphic R-factors ⇔ there exists a primitive idempotent g such that e₁Rg 6= 0 and e₂Rg6= 0.

Alt. Def. Two primitive idempotents eand f are linked if there exist primitive idempotents

e=e1, e2, . . . , en =f such that e_iRe_i+1 6= 0 or e_i+1Re_i 6= 0 for eachi.

Fix a PPO-set W ={e₁, . . . , e_n}of R.

LetW₁, . . . , W_r be equivalence classes ofW under linkage.

Theorem 7. R is uniquely expressible as R=B₁⊕ · · · ⊕B_t where each B_i is an indecomposable ideal of R.

The ideals Bi are called the blocks of R.

Theorem 8. If R is tame, r = t and the ideals generated by the equivalence classes W_i are the same as the blocks of R.

AnR-moduleM isblock indecomposableifM cannot be written as a direct sumM₁⊕M₂ whereM₁ andM₂ have no isomorphicR-factors.

Theorem 9. If G is a faithful tame R-module, then G is uniquely expressible as

G=G₁⊕. . .⊕G_t

where each Gi is a block indecomposable R-ideal of G and Gi =GBi. The R-idealsG_i are called the blocks of G.

THE IDEMPOTENT QUIVER OF A NEARRING 3

Theorem 10. For ei, ej ∈W, TFAE:

1. e_iR 'e_jR.

2. e_iR 'e_jR where R=R/J₂(R).

3. e_ire_j is not in J₂(R) for some r ∈R−J₂(R).

4. e_iRe_jR contains a primitive idempotent.

Let e_i ∼ e_j if one of 1-4 of Theorem 10 holds. Note that if e_i ∼ e_j, then e_i and e_j are linked.

Choose a set of representatives

V ={e₁, . . . , e_m}

(relabeling if necessary) of the equivalence classes of W under∼.

The quiver of R, denoted Γ(R), is the directed graph with vertex set V and directed edges formed by drawing an arrow from e_i to e_j if eiRej 6= 0.

Theorem 11. e_iRe_j 6= 0 ⇔e_iJ₂(R)e_j 6= 0.

Theorem 12. IfW⁰ is another PPO-set ofRand ifV⁰ is a set of equiv- alence class representatives of W⁰ under∼, then the quivers formed by V and V⁰ are isomorphic.

Theorem 13. If R is tame and G is a faithful tame R-module, then the connected components of Γ(R) are in one-to-one correspondence with the blocks of R and G.

Suppose G is a faithful tameR-module.

The socle of G, Soc(G), is the sum of the minimal R-ideals of G.

The socle series of G is

0≤Soc₁(G)≤Soc₂(G)≤. . .

where Soc₁(G) = Soc(G) and Soc_i+1(G)/Soc_i(G) = Soc(G/Soc_i(G)).

Theorem 14. There exists an n such that Socn(G) = G and

Soci+1(G)/Soci(G) is a direct sum of type 2 R-modules. Also, if e is a primitive idempotent and R = R/J₂(R), eR is isomorphic to a summand of Soc_i+1(G)/Soc_i(G) for some i.

Theorem 15. Suppose H < K < L areR-ideals of G such that K/H and L/K are nonisomorphic type 2 R-modules. Let e, f ∈V such that

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eR ' L/K and f R ' K/H. If L/H is an indecomposable R-module, then Γ(R) contains an arrow from e to f

Theorem 16. Suppose e ∈ V. If H is a type 2 summand of Soc(G) such that eR'H and G/Soc(G) contains no factor isomorphic to H, then e cannot be an initial vertex of an arrow of Γ(R).

Using SONATA to Calculate Quivers 1. Find the set of nonzero idempotentsI of R.

2. Use the definition of primitive idempotent to find the set of primitive idempotents P from I.

3. Filter a PPO-set W fromP as follows:

(i) Choose an element e1 of P. If

Ann_R(e₁) ∩ P = ∅, done. If not, choose an element e₂ in Ann_R(e₁)∩P.

(ii) If e₂e₁ = 0, go to (iii). If not, let f = (e1−e2e1)e1. Choose f₁ ∈f R∩P.

Let e₁ =f₁f.

(iii) Consider AnnR(e1, e2)∩P . . . 4. To find V, proceed through

e₁, e₂, . . . , e_n, deleting e_j using one of the following approaches:

(i) if for some i < j we have e_iR 'e_jR, (ii) if for some i < j we have e_iR 'e_jR,

(iii) if for some i < j we have e_ire_j is not in J₂(R) for some r ∈ R−J2(R)

(iv) if for some i < j we have eiRejR∩P 6=∅.

5. Determine the set of directed edges E of Γ(R) by having an arrow from e_i to e_j whenever e_iRe_j 6= 0 (or e_iJ₂(R)e_j 6= 0) and draw the quiver Γ(R).