ON ARMENDARIZ-LIKE PROPERTIES

2014

We introduce the notion of J-Armendariz rings, which are a generalization of weak Armendariz rings and investigate their properties. We show that any local ring is J-Armendariz, and then find a local ring that is not weak Armendariz. Moreover, we prove that a ring R is J-Armendariz if and only if the n-by-n upper triangular matrix ring Tn(R) is J-Armendariz. For a ring R and for some e 2 = e ∈ R, we show that if R is an abelian ring, then R is J-Armendariz if and only if eRe is J-Armendariz.

2010

Let R be a ring, S a strictly ordered monoid, and ω : S → End(R) a monoid homomorphism. The skew generalized power series ring R[ [S, ω]] is a common generalization of (skew) polynomial rings, (skew) power series rings, (skew) Laurent polynomial rings, (skew) group rings, and Mal'cev-Neumann Laurent series rings. We study the (S, ω)-Armendariz condition on R, a generalization of the standard Armendariz condition from polynomials to skew generalized power series. We resolve the structure of (S, ω)-Armendariz rings and obtain various necessary or sufficient conditions for a ring to be (S, ω)-Armendariz, unifying and generalizing a number of known Armendariz-like conditions in the aforementioned special cases. As particular cases of our general results we obtain several new theorems on the Armendariz condition; for example, left uniserial rings are Armendariz. We also characterize when a skew generalized power series ring is reduced or semicommutative, and we obtain partial characterizations for it to be reversible or 2-primal.

2014

Journal of Algebra, 2000

Journal of Pure and Applied Algebra, 2003

A ring R is called reversible if ab = 0 implies ba = 0 for a; b ∈ R. We continue in this paper the study of reversible rings by Cohn [4]. We ÿrst consider properties and basic extensions of reversible rings and related concepts to reversible rings, including some kinds of examples needed in the process. We next show that polynomial rings over reversible rings need not be reversible, and sequentially argue about the reversibility of some kinds of polynomial rings. Moreover we prove that if R is a reduced ring then R[x]=(x n ) is a reversible ring, where (x n ) is the ideal generated by x n and n is a positive integer; and that for a right Ore ring R with Q its classical right quotient ring, R is reversible if and only if Q is reversible.

2014

In the present paper we concentrate on a natural generalization of NC-McCoy rings that is called J-McCoy and investigate their properties. We prove that local rings are J-McCoy. Also, for an abelian ring R, we show that R is J-McCoy if and only if eR is J-McCoy, where e is an idempotent element of R. Moreover, we give an example to show that the J-McCoy property does not pass Mn(R), but S(R, n), A(R, n), B(R, n) and T (R, n) are J-McCoy.

ON THE ANDERSON-BADAWI ω R [X] (I[X ]) = ω R (I) CONJECTURE PEYMAN NASEHPOUR ABSTRACT. Let R be a commutative ring with identity and n be a positive integer. Anderson and Badawi, in their paper on n-absorbing ideals, define a proper ideal I of a commutative ring R to be an n-absorbing ideal of R, if whenever x 1 · · · x n+1 ∈ I for x 1 , · · · , x n+1 ∈ R, then there are n of the x i 's whose product is in I and conjecture that ω R[X] (I[X]) = ω R (I) for any ideal I of an arbitrary ring R, where ω R (I) = min{n : I is an n-absorbing ideal of R}. In the present paper, we use content formula techniques to prove that their conjecture is true, if one of the following conditions hold:

Monatshefte Fur Mathematik, 2011

A ring is called CT (commutative transitive) if commutativity is a transitive relation on its nonzero elements. Likewise, it is wCT (weakly commutative transitive) if commutativity is a transitive relation on its noncentral elements. The main topic of this paper is to describe the structure of finite wCT rings. It is shown that every such ring is a direct sum of an indecomposable noncommutative wCT ring of prime power order, and a commutative ring. Furthermore, finite indecomposable wCT rings are either two-by-two matrices over fields, local rings, or basic rings with two maximal ideals. We characterize finite local rings as generalized skew polynomial rings over coefficient Galois rings; the associated automorphisms of the Galois ring give rise to a signature of the local ring. These are then used to further describe the structure of finite local and wCT basic rings.

2012

Let R be a commutative ring with nonzero identity. In this paper, we study the von Neumann regular elements of R. We also study the idempotent elements, π-regular elements, the von Neumann local elements, and the clean elements of R. Finally, we investigate the subgraphs of the zero-divisor graph Γ(R) of R induced by the above elements.

2014