Let R be a commutative ring with identity and A(R) be the set of ideals with nonzero annihilator. The annihilating-ideal graph of R is defined as the graph AG(R) with the vertex set A(R) = A(R)r {0} and two distinct vertices I and J are adjacent if and only if IJ = 0. In this paper, we study the domination number of AG(R) and some connections between the domination numbers of annihilating-ideal...

let $r$ be a commutative noetherian ring with non-zero identity and $fa$ an ideal of $r$. let $m$ be a finite $r$--module of finite projective dimension and $n$ an arbitrary finite $r$--module. we characterize the membership of the generalized local cohomology modules $lc^{i}_{fa}(m,n)$ in certain serre subcategories of the category of modules from upper bounds. we define and study the properti...

we show that the lattice of all ideals of a ring $r$ can be embedded in the lattice of all its fuzzyideals in uncountably many ways. for this purpose, we introduce the concept of the generalizedcharacteristic function $chi _{s}^{r} (a)$ of a subset $a$ of a ring $r$ forfixed $r , sin [0,1] $ and show that $a$ is an ideal of $r$ if, and only if, its generalizedcharacteristic function $chi _{s}^{...

A finitely generated $R$-module is said to be a module of type ($F_r$) if its $(r-1)$-th Fitting ideal is the zero ideal and its $r$-th Fitting ideal is a regular ideal. Let $R$ be a commutative ring and $N$ be a submodule of  $R^n$ which is generated by columns of  a matrix $A=(a_{ij})$ with $a_{ij}in R$ for all $1leq ileq n$, $jin Lambda$, where $Lambda $ is a (possibly infinite) index set.  ...

We show that the existence of a nontrivial proper ideal in a commutative ring with identity which is not a eld is equivalent to WKL0 over RCA0, and that the existence of a nontrivial proper nitely generated ideal in a commutative ring with identity which is not a eld is equivalent to ACA0 over RCA0. We also prove that there are computable commutative rings with identity where the nilradical is ...

Let R be a commutative ring with identity and let I two-generated ideal of R. We denote by SL2(R) the group 2×2 matrices over determinant 1. study action matrix right-multiplication on V2(I), set generating pairs I. Fitt1(I) second Fitting Our main result asserts that V2(I)/SL2(R) identifies units R/Fitt1(I) via natural generalization if can generated two regular elements. This is illustrated i...