Let F be a field of characteristic two and G a finite abelian 2-group with an involutory automorphism η. If G = H × D with non-trivial subgroups H and D of G such that η inverts the elements of H (H without a direct factor of order 2) and fixes D element-wise, then the linear extension of η to the group algebra FG is called a nice involution. This determines the groups of unitary and symmetric normalized units of FG. We calculate the orders and the invariants of these subgroups.

Let $R$ be a commutative ring with identity. The co-annihilating-ideal graph of $R$, denoted by ${{\mathcal{A}}_{R}}$, is a graph whose vertex set is the set of all non-zero proper ideals of $R$ and two distinct vertices $I$ and $J$ are adjacent whenever $\text{Ann}\left( I \right)\,\cap \,\text{Ann}\left( J \right)\,=\,\left\{ 0 \right\}$. In this paper we initiate the study of the co-annihilating ideal graph of a commutative ring and we investigate its properties.

Let R and S be commutative rings, not necessarily with identity. We investigate the ideals, prime ideals, radical ideals, primary ideals, and maximal ideals of R×S. Unlike the case where R and S have an identity, an ideal (or primary ideal, or maximal ideal) of R×S need not be a ‘subproduct’ I×J of ideals. We show that for a ring R, for each commutative ring S every ideal (or primary ideal, or maximal ideal) is a subproduct if and only if R is an e-ring (that is, for r∈R, there exists er∈R with err=r) (or u-ring (that is, for each proper ideal A of R, )), the Abelian group (R/R2 ,+) has no maximal subgroups).