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...

We recall that a ring R is called near pseudo-valuation ring if every minimal prime ideal is a strongly prime ideal. Let R be a commutative ring, σ an automorphism of R. Recall that a prime ideal P of R is σ-divided if it is comparable (under inclusion) to every σ-stable ideal I of R. A ring R is called a σ-divided ring if every prime ideal of R is σ-divided. Also a ring R is almost σ-divided r...