A generalization of anti-commutative rings arising from 2-varieties

Let R be a non-associative ring of characteristic not 2 or 3 which satisfies the identities (ab+ba)c = (ac+ca)b, a(bc+cb) = b(ac+ca), and a².a = a.a². It is proved that R is power asso-ciative, that if R is simple, then R is either anti-commutative or else commutative and associative. It is shown that if R is nil and semiprime, then R is anti-commutative, and an example of a prime ring of this type which is neither commutative nor anti-commu-tative is given.     

A generalization of anti-commutative rings arising from 2-varieties

Let Rbe a nonassociative ring of characteristic not 2 or 3 which satisfies the identities (ab=ba) = (ac+ca)b, a(ac+ca) = b(ac+ca) and a²a = aa². We show that these rings are characterized as associative commutative rings with a type of biadditive mapping. From this characterization we show easily that simple rings are associative-commutative, or anti-commutative. Among the examples given, is a finite dimensional algebra which is solvable but not nilpotent.     

A generalization of zero divisor graphs associated to commutative rings

Let R be a commutative ring with a nonzero identity element. For a natural number n, we associate a simple graph, denoted by \(\Gamma ^n_R\), with \(R^n\backslash \{0\}\) as the vertex set and two distinct vertices X and Y in \(R^n\) being adjacent if and only if there exists an \(n\times n\) lower triangular matrix A over R whose entries on the main diagonal are nonzero and one of the entries on the main diagonal is regular such that \(X^TAY=0\) or \(Y^TAX=0\), where, for a matrix \(B, B^T\) is the matrix transpose of B. If \(n=1\), then \(\Gamma ^n_R\) is isomorphic to the zero divisor graph \(\Gamma (R)\), and so \(\Gamma ^n_R\) is a generalization of \(\Gamma (R)\) which is called a generalized zero divisor graph of R. In this paper, we study some basic properties of \(\Gamma ^n_ R\). We also determine all isomorphic classes of finite commutative rings whose generalized zero divisor graphs have genus at most three.     

A note on semiprime right Goldie rings

It is shown that a ring R is semiprime right Goldie if and only if R is right nonsingular and every nonsingular right R-module M has a direct decomposition M = I⊕N, where I is injective and N is a reduced module such that N does not contain any extending submodule of infinite Goldie dimension.     

On right G-semilocal rings

The notion of semilocal ring is extended to the classes of right G-semilocal and right N-semilocal rings. We explore the algebraic properties of such classes and study their relations with many other rings such as clean, exchange and I-finite rings. Localization of right G-semilocal rings is considered.     

Artinian right serial rings

Let be an artinian ring such that for the Jacobson radical of , is a direct product of matrix rings over finite-dimensional division rings. Then the following are proved to be equivalent: (1) Every indecomposable injective left -module is uniserial. (2) is right serial.

     

A class of loops with right alternative loop rings

The purpose of this paper is to exhibit a class of loops which have strongly right alternative loop rings that are not alternative..     

csp-rings as a generalization of rings of pseudo-rational numbers

Ideals and factor rings of the so-called csp-rings are described, and modules over such rings are considered. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 13, No. 3, pp. 35–38, 2007.     

Left and right distributive rings

By a distributive module we mean a module with a distributive lattice of submodules. LetA be a right distributive ring that is algebraic over its center and letB be the quotient ring ofA by its prime radicalH. ThenB is a left distributive ring, andH coincides with the set of all nilpotent elements ofA.Translated fromMatematicheskie Zametki, Vol. 58, No. 4, pp. 604–627, October, 1995.     

On flat factor rings and fully right idempotent rings

Summary We characterize those two-sided idealsf of a ringR such thatR/f is flat as a right or leftR-module. Furthermore we study those ringsR such thatR/f is leftR-flat for each two-sided ideal f; in particular whenR has finite Goldie dimension.

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Riassunto Vengono caratterizzati gli ideali bilateri f di un anelloR tali cheR/f è piatto comeR-modulo sinistro o destro. Vengono inoltre studiati gli anelliR tali che per ogni ideale bilaterof l'R-modulo sinistroR/f è piatto, con particolare riguardo al caso in cuiR ha dimensione di Goldie finita.

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Lavoro eseguito nell'ambito del G.N.S.A.G.A. del Consiglio Nazionale delle Ricerche.     

On nonsingular right fpf rings

A right FPF ring is one over which every finitely generated faithful right module is a generator. The main purpose of the article is to givp the following cnaracterization of certain right FPF rings. TheoremLet R be semiprime and right semihereditary. Then R is right FPF iff (1) the right maximal ring of quotients Q_r (R) = Q coincides with the left and right classical rings of quotients and is self-injective regular of bounded index, (ii) R and Q have the same central idem-potents, (iii) if I is an ideal of R generated by a ma­ximal ideal of the boolean algebra of central idempotent s5 R/I is such that each non-zero finitely generated right ideal is a generator (hence prime), and (iv) R is such that every essential right ideal contains an ideal which is essential as a right ideal

In case that R is semiprime and module finite over its centre C, then the above can be used to show that R is FPF (both sides) if and only if it is a semi-hereditary maximal C-order in a self-injective regular ring (of finite index)

In order to prove the above it is shown that for any semiprime right FPF ring R, Q _lcl (R) exists and coincides with Q_r(R) (Faith and Page have shown that the latter is self-injective regular of bounded index). It R is semiprime right FPF and satisfies a polynamical identity then the factor rings as in (iii) above are right FPF and R is the ring of sections of a sheaf of prime right FPF rings

The Proofs use many results of C. Faith and S Page as well as some of the techniques of Pierce sheaves