On Nil-Symmetric Rings

The concept of nil-symmetric rings has been introduced as a generalization of symmetric rings and a particular case of nil-semicommutative rings. A ring is called right (left) nil-symmetric if, for , where are nilpotent elements, implies . A ring is called nil-symmetric if it is both right and left nil-symmetric. It has been shown that the polynomial ring over a nil-symmetric ring may not be a right or a left nil-symmetric ring. Further, it is also proved that if is right (left) nil-symmetric, then the polynomial ring is a nil-Armendariz ring. 1. Introduction Throughout this paper, all rings are associative with unity. Given a ring , and denote the set of all nilpotent elements of and the polynomial ring over , respectively. A ring is called reduced if it has no nonzero nilpotent elements; is said to be Abelian if all idempotents of are central; is symmetric [1] if implies for all . An equivalent condition for a ring to be symmetric is that whenever product of any number of elements of the ring is zero, any permutation of the factors still gives the product zero [2]. is reversible [3] if implies for all ; is called semicommutative [4] if implies for all . In [5], Rege-Chhawchharia introduced the concept of an Armendariz ring. A ring is called Armendariz if whenever polynomials , satisfy , then for each . Liu-Zhao [6] and Antoine [7] further generalize the concept of an Armendariz ring by defining a weak-Armendariz and a nil-Armendariz ring, respectively. A ring is called weak-Armendariz if whenever polynomials ,？？ satisfy , then for each . A ring is called nil-Armendariz if whenever ,？？ satisfy , then for each . Mohammadi et al. [8] initiated the notion of a nil-semicommutative ring as a generalization of a semicommutative ring. A ring is nil-semicommutative if implies for all . In their paper it is shown that, in a nil-semicommutative ring , forms an ideal of . Getting motivated by their paper we introduce the concept of a right (left) nil-symmetric ring which is a generalization of symmetric rings and a particular case of nil-semicommutative rings. Thus all the results valid for nil-semicommutative rings are valid for right (left) nil-symmetric rings also. We also prove that if a ring is right (left) nil-symmetric and Armendariz, then is right (left) nil-symmetric. In the context, there are also several other generalizations of symmetric rings (see [9, 10]). 2. Right (Left) Nil-Symmetric Rings For a ring , and denote the full matrix ring and the upper triangular matrix ring over , respectively. We observe that if is a ring, then Definition 1. A ring