Narrow Results

The notions of inner-star and star-inner generalized inverses are introduced on the set of all regular elements in a ∗-ring ${ℛ}$. We thus extend the concept of inner-star and star-inner complex matrices. We study properties of these hybrid generalized inverses on the set ${{ℛ}}^{†}$ of all Moore–Penrose invertible elements in ${ℛ}$ and thus generalize some known results. Partial orders that are induced by inner-star and star-inner inverses are introduced on ${{ℛ}}^{†}$, their properties are examined, and their characterizations are presented.

Results of Davis on normal pairs (R, T) of domains are generalized to (commutative) rings with nontrivial zero-divisors, particularly complemented rings. For instance, if T is a ring extension of an almost quasilocal complemented ring R, then (R, T) is a normal pair if and only if there is a prime ideal P of R such that T = R_[P], R/P is a valuation domain and PT = P. Examples include sufficient conditions for the "normal pair" property to be stable under formation of infinite products and ⋈ constructions.

If $A\subseteq B$ are (commutative) rings, $[A,B]$ denotes the set of intermediate rings and $(A,B)$ is called an almost valuation (AV)-ring pair if each element of $[A,B]$ is an AV-ring. Many results on AV-domains and their pairs are generalized to the ring-theoretic setting. Let $R\subseteq S$ be rings, with ${\overline{R}}_S$ denoting the integral closure of $R$ in $S$. Then $(R,S)$ is an AV-ring pair if and only if both $(R,{\overline{R}}_S)$ and $({\overline{R}}_S,S)$ are AV-ring pairs. Characterizations are given for the AV-ring pairs arising from integrally closed (respectively, integral; respectively, minimal) ring extensions $R\subseteq S$. If $(R,S)$ is an AV-ring pair, then $R\subseteq S$ is a P-extension. The AV-ring pairs $(R,S)$ arising from root extensions are studied extensively. Transfer results for the “AV-ring” property are obtained for pullbacks of $(B,I,D)$ type, with applications to pseudo-valuation domains, integral minimal ring extensions, and integrally closed maximal non-AV subrings. Several sufficient conditions are given for $(R,S)$ being an AV-ring pair to entail that $S$ is an overring of $R$, but there exist domain-theoretic counter-examples to such a conclusion in general. If $(R,S)$ is an AV-ring pair and $R\subseteq S$ satisfies FCP, then each intermediate ring either contains or is contained in ${\overline{R}}_S$. While all AV-rings are quasi-local going-down rings, examples in positive characteristic show that an AV-domain need not be a divided domain or a universally going-down domain.

We give an example to show that, for nonunital rings $R$, the direct sum $aR\oplus bR=(a+b)R$ with $a+b$ regular has no in general right-left symmetry. It is then proved that the right-left symmetry actually holds in a left and right faithful ring.

Let $R$ be a commutative ring and let $N$ be a submodule of $R^n$ which consists of columns of a matrix $A=(a_{ij})$ with $a_{ij}\in R$ for all $1\le i\le n$, $j\in \mathrm{\Lambda }$, where $\mathrm{\Lambda }$ is an index set. For every $\mu =\{j_1,\dots ,j_q\}\subseteq \mathrm{\Lambda }$, let I${}_{\mu }(N)$ be the ideal generated by subdeterminants of size $q$ of the matrix $(a_{ij}:1\le i\le n,j\in \mu )$. Let $M=R^n/N$. In this paper, we obtain a constructive description of T($M$) and we show that when $R$ is a local ring, $M$/T($M$) is free of rank $n-q$ if and only if I${}_{\mu }(N)$ is a principal regular ideal, for some $\mu =\{j_1,\dots ,j_q\}\subseteq \mathrm{\Lambda }$. This improves a lemma of Lipman which asserts that, if I($M$) is the $(m-q)$th Fitting ideal of $M$ then I($M$) is a regular principal ideal if and only if $N$ is finitely generated free and $M$/T($M$) is free of rank $m-q.$

Let $T(X)$ be the full transformation monoid on a nonempty set $X$. An element $f$ of $T(X)$ is said to be semi-balanced if the collapse of $f$ is equal to the defect of $f$. In this paper, we prove that an element of $T(X)$ is unit-regular if and only if it is semi-balanced. For a partition ${𝒫}$ of $X$, we characterize unit-regular elements in the monoid $T(X,{𝒫})=\{f\in T(X)|(\forall X_i\in {𝒫})(\exists X_j\in {𝒫})X_{i}f\subseteq X_j\}$ under composition. We characterize regular elements in the submonoids $\mathrm{\Sigma }(X,{𝒫})=\{f\in T(X,{𝒫})|(\forall X_i\in {𝒫})Xf\cap X_i\ne \varnothing \}$ and $\mathrm{\Omega }(X,{𝒫})=\{f\in T(X,{𝒫})|(\forall x,y\in X,x\ne y)(x,y)\in E\Rightarrow xf\ne yf\}$ of $T(X,{𝒫})$, where $E$ is the equivalence induced by ${𝒫}$. We also characterize unit-regular elements in $\mathrm{\Sigma }(X,{𝒫})$, $\mathrm{\Omega }(X,{𝒫})$, and the other two known submonoids of $T(X,{𝒫})$.

The aim of the paper is to introduce a new class of rings — the inner generalized Weyl algebras (IGWA) — and to give simplicity criteria for them. For each IGWA $A$ a derivative series of IGWAs, $A\to A^{\prime }\to A^{″}\to \cdots \to A^{(\alpha )}\to \cdots$, is attached where $\alpha$ is an arbitrary ordinal. In general, all rings $A^{(\alpha )}$ are distinct. A new construction of rings, the inner $(\sigma ,\tau ,a)$-extension of a ring, is introduced (where $\sigma$ and $\tau$ are endomorphisms of a ring $D$ and $a\in D$).

The object of this article is to initiate the study of a class of rings in which the right duo property is applied in relation to powers of elements and the monoid of all regular elements. Such rings shall be called right exp-DR. We investigate the structures of group rings, right quotient rings, matrix rings and (skew) polynomial rings, through the study of right exp-DR rings. In addition, we provide a method of constructing finite non-abelian $p$-groups for any prime $p$.

Let $R$ be a commutative ring, $f\in R[X_1,\dots ,X_k]$ a multivariate polynomial, and $G$ a finite subgroup of the group of units of $R$ satisfying a certain constraint, which always holds if $R$ is a field. Then, we evaluate $\sum f(x_1,\dots ,x_k)$, where the summation is taken over all pairwise distinct $x_1,\dots ,x_k\in G$. In particular, let $p^s$ be a power of an odd prime, $n$ a positive integer coprime with $p-1$, and $a_1,\dots ,a_k$ integers such that $\phi (p^s)$ divides $a_1+\cdots +a_k$ and $p-1$ does not divide ${\sum }_{i\in I}{a}_i$ for all non-empty proper subsets $I\subseteq \{1,\dots ,k\}$; then