let $r$ be a 2-torsion free ring and $u$ be a square closed lie ideal of $r$. suppose that $alpha, beta$ are automorphisms of $r$. an additive mapping $delta: r longrightarrow r$ is said to be a jordan left $(alpha,beta)$-derivation of $r$ if $delta(x^2)=alpha(x)delta(x)+beta(x)delta(x)$ holds for all $xin r$. in this paper it is established that if $r$ admits an additive mapping $g : rlongrigh...

In an algebra class, one uses the zero-factor property to solve polynomial equations. For example, consider the equation x 2 = x. Rewriting it as x (x − 1) = 0, we conclude that the solutions are x = 0, 1. However, the same equation in a different number system need not yield the same solutions. For example, in Z 6 (the integers modulo 6), not only 0 and 1, but also 3 and 4 are solutions. (Chec...

It is proved that for a ring R either an affine algebra over field, or equicharacteristic complete local ring, some power of the Jacobian ideal annihilates ExtRd+1(−,−), where d Krull dimension R. Sufficient conditions are identified under which itself these Ext-modules, and examples provided show this not always case. The crucial new idea to consider derived version Noether different algebra.

Let $I$ be a proper ideal of a commutative semiring $R$ and let $P(I)$ be the set of all elements of $R$ that are not prime to $I$. In this paper, we investigate the total graph of $R$ with respect to $I$, denoted by $T(Gamma_{I} (R))$. It is the (undirected) graph with elements of $R$ as vertices, and for distinct $x, y in R$, the vertices $x$ and $y$ are adjacent if and only if $x + y in P(I)...

let  $r$ be a commutative noetherian ring and $i$ be an ideal of $r$. we say that $i$ satisfies the persistence property if  $mathrm{ass}_r(r/i^k)subseteq mathrm{ass}_r(r/i^{k+1})$ for all positive integers $kgeq 1$, which $mathrm{ass}_r(r/i)$ denotes the set of associated prime ideals of $i$. in this paper, we introduce a class of square-free monomial ideals in the polynomial ring  $r=k[x_1,ld...