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.  ...

In this paper, we introduce generalized adjoint semigroups (GA-semigroups) of a ring R. We construct generalized adjoint semigroups on a ring R by means of bitranslations of R. It is shown that GA-semigroups of a π-regular ring are π-regular. As an application we deduce that in any ring, idempotents can be lifted modulo π-regular ideals. GA-semigroups containing idempotents are described in ter...

We study finiteness conditions on large tilting modules over arbitrary rings. We then turn to a hereditary artin algebra R and apply our results to the (infinite dimensional) tilting module L that generates all modules without preprojective direct summands. We show that the behaviour of L over its endomorphism ring determines the representation type of R. A similar result holds true for the (in...

We study the first order theory of Bezout difference rings. In particular we show that rings of sequences very rarely have decidable theories as difference rings, or even decidable model completions. MSC: 03C60, 03B25, 12H10, 13L05

Ayache has recently proved that if R is an integrally closed domain such that each overring of R is treed, then R is a locally pseudo-valuation domain. We investigate the extent to which the analogue (in which one concludes that R is a locally pseudo-valuation ring) holds if R is generalized to a commutative ring. A positive result is obtained if R is an idealization D(+)K where D is an integra...

It is proved that radical of every generalized ring is a G-graded ideal when its index is an abelian group G. The relation between radicals of generalized rings and Γ-rings is given. The explicit formulas for generalized ring A and radical properties r = rb, rl, rj , rn are obtained: r(A) = g.m.r(A) = ∑ {r(Aij) | i, j ∈ I}. The relation between the radicals of path algebras and connectivity of ...

We give a direct, self-contained proof of the fact that for a large class of rings A , in particular for all regular rings with involution, W (A[t, 1/t]) = W (A)⊕W (A) .

In this paper we define a new type of rings ”almost powerhermitian rings” (a generalization of almost hermitian rings) and establish several sufficient conditions over a ring R such that, every regular matrix admits a diagonal power-reduction.

Recall that a regular local ring is a noetherian local ring with dimension equal to dimkm/m. A regular local ring of dimension one is precisely a discrete valuation ring. If V is a nosingular variety then t(V ) is a scheme with all local rings regular, so t(V ) is clearly regular in codimension one. In this section we will consider schemes satisfying the following condition: (∗) X is a noetheri...