Journal of Algebra and Related Topics
Volume:3 Issue: 2, Autumn 2015

In this paper, we introduce a class of J-quasipolar rings. Let R be a ring with identity. An element a of a ring R is called {\it weakly J-quasipolar} if there exists p2=p∈comm2(a) such that a or a−p are contained in J(R) and the ring R is called {\it weakly J-quasipolar} if every element of R is weakly J-quasipolar. We give many characterizations and investigate general properties of weakly J-quasipolar rings. If R is a weakly J-quasipolar ring, then we show that (1) R/J(R) is weakly J-quasipolar, (2) R/J(R) is commutative, (3) R/J(R) is reduced. We use weakly J-quasipolar rings to obtain more results for J-quasipolar rings. We prove that the class of weakly J-quasipolar rings lies between the class of J-quasipolar rings and the class of quasipolar rings. Among others it is shown that a ring R is abelian weakly J-quasipolar if and only if R
is uniquely clean.

In this paper the concept of an Ω- Almost Boolean ring is introduced and illistrated how a sheaf of algebras can be constructed from an Ω- Almost Boolean ring over a locally Boolean space.

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(ΓI(R)). It is the (undirected) graph with elements of R as vertices, and for distinct x,y∈R, the vertices x and y are adjacent if and only if x∈P(I). The properties and possible structures of the two (induced) subgraphs P(ΓI(R)) and P¯(ΓI(R)) of T(ΓI(R)), with vertices P(I) and R−P(I), respectively are studied.

Let R be an arbitrary ring and T be a submodule of an R-module M. A submodule N of M is called T-small in M provided for each submodule X of M, T⊆X implies that T⊆X. We study this mentioned notion which is a generalization of the small submodules and we obtain some related results.

Let R=k[x1,x2,⋯,xN] be a polynomial ring over a field k. We prove that for any positive integers m,n, reg(ImJnK)≤mreg(I)麷(J)귨(K) if I,J,K⊆R are three monomial complete intersections (I, J, K are not necessarily proper ideals of the polynomial ring R), and I,J are of the form (xa1i1,xa2i2,⋯,xalil).

Here we consider all finite non-abelian 2-generator p-groups (p an odd prime) of nilpotency class two and study the probability of having nth-roots of them. Also we find integers n for which, these groups are n-central.