Related papers: When is a subgroup of a ring an ideal?

In this paper, we study the question of when the symmetric units in an integral group ring ZG form a multiplicative group. When G is periodic, necessary and sufficient conditions are given for this to occur.

In this note, we define and investigate ideal covering numbers of associative rings (not assumed to be commutative or unital): three invariants defined as the minimal number of proper left, right, or two-sided ideals whose union equals the…

There is a longstanding conjecture by Fr\"oberg about the Hilbert series of the ring $R/I$, where $R$ is a polynomial ring, and $I$ an ideal generated by generic forms. We prove this conjecture true in the case when $I$ is generated by a…

In this paper using the connections between some subvarieties of residuated lattices, we investigated some properties of the lattice of ideals in commutative and unitary rings. We give new characterizations for commutative rings $A$ in…

In the present paper we continue the study of the elementary commutator subgroups $[E(n,A),E(n,B)]$, where $A$ and $B$ are two-sided ideals of an associative ring $R$, $n\ge 3$. First, we refine and expand a number of the auxiliary results,…

Let $(R,\mathfrak{m},k)$ be a Noetherian local ring and let $M$ be a finitely generated $R$-module. The main focus of this paper is to give positive answers for some long-standing homological conjectures over the idealization ring $R\ltimes…

For a graded domain $R=k[X_0,...,X_m]/J$ over an arbitrary domain $k$, it is shown that the ideal generated by elements of degree $\geq mA$, where $A$ is the least common multiple of the weights of the $X_i$, is a normal ideal.

In this paper, we introduce $\phi$-1-absorbing prime ideals in commutative rings. Let $R$ be a commutative ring with a nonzero identity $1\neq0$ and $\phi:\mathcal{I}(R)\rightarrow\mathcal{I}(R)\cup\{\emptyset\}$ be a function where…

All rings are commutative with $1$ and $n$ is a positive integer. Let $\phi: J(R)\to J(R)\cup{\emptyset}$ be a function where $J(R)$ denotes the set of all ideals of $R$. We say that a proper ideal $I$ of $R$ is $\phi$-$n$-absorbing primary…

We investigate ideal-semisimple and congruence-semisimple semirings. We give several new characterizations of such semirings using e-projective and e-injective semimodules. We extend several characterizations of semisimple rings to (not…

Let $R$ be a commutative ring. An $R$-module $M$ is said to be almost projective if ${\rm Ext}^1_R(M, N) = 0$ for any $R_{\mathfrak{m}}$-module $N$ and any maximal ideal $\mathfrak{m}$ of $R$. In this paper, we investigate rings $R$ over…

We extend the notion of standard pairs to the context of monomial ideals in semigroup rings. Standard pairs can be used as a data structure to encode such monomial ideals, providing an alternative to generating sets that is well suited to…

Let K be a compact Lie group and W a finite-dimensional real K-module. Let X be a K-stable real algebraic subset of W. Let I(X) denote the ideal of X in R[W] and let I_K(X) be the ideal generated by I(X)^K. We find necessary conditions and…

Let $R$ be a maximal subring of a ring $T$, and $(R:T)$, $(R:T)_l$ and $(R:T)_r$ denote the greatest ideal, left ideal and right ideal of $T$ which are contained in $R$, respectively. It is shown that $(R:T)_l$ and $(R:T)_r$ are prime…

In this paper, we introduce a concept of $\mathfrak{X}$-element with respect to an $M$-closed set $\mathfrak{X}$ in multiplicative lattices and study properties of $\mathfrak{X}$-elements. For a particular $M$-closed subset $\mathfrak{X}$,…

Let $G$ be a group, $R$ be a $G$-graded commutative ring with nonzero unity and $GI(R)$ be the set of all graded ideals of $R$. Suppose that $\phi:GI(R)\rightarrow GI(R)\cup\{\emptyset\}$ is a function. In this article, we introduce and…

In this article we investigate when a homogeneous ideal in a graded ring is normal, that is, when all positive powers of the ideal are integrally closed. We are particularly interested in homogeneous ideals in an N-graded ring generated by…

Let $(\FormR)$ be a form ring such that $A$ is quasi-finite $R$-algebra (i.e., a direct limit of module finite algebras) with identity. We consider the hyperbolic Bak's unitary groups $\GU(2n,\FormR)$, $n\ge 3$. For a form ideal…

Let $R$ be a ring with unity. The graph $\Gamma(R)$ is a graph with vertices as elements of $R$, where two distinct vertices $a$ and $b$ are adjacent if and only if $Ra+Rb=R$. Let $\Gamma_2(R)$ is the subgraph of $\Gamma(R)$ induced by the…

We characterize the commutative rings whose ideals (resp. regular ideals) are products of radical ideals.