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Let T_n be the full transformation semigroup of a finite set, say X_n = {1, 2, …, n}, and for a given full transformation α : X_n → X_n let F(α) = {x ∈ X_n : xα = x} be its set of fixed points. In this note we obtain and discuss formulae for F(n, r, k) = |{α ∈ T_n : |Im α| = r ∧ |F(α)| = k}|.

Let 𝒫_X and 𝒮_X be the partition monoid and symmetric group on an infinite set X. We show that 𝒫_X may be generated by 𝒮_X together with two (but no fewer) additional partitions, and we classify the pairs α, β ∈ 𝒫_X for which 𝒫_X is generated by 𝒮_X ∪ {α, β}. We also show that 𝒫_X may be generated by the set ℰ_X of all idempotent partitions together with two (but no fewer) additional partitions. In fact, 𝒫_X is generated by ℰ_X ∪ {α, β} if and only if it is generated by ℰ_X ∪ 𝒮_X ∪ {α, β}. We also classify the pairs α, β ∈ 𝒫_X for which 𝒫_X is generated by ℰ_X ∪ {α, β}. Among other results, we show that any countable subset of 𝒫_X is contained in a 4-generated subsemigroup of 𝒫_X, and that the length function on 𝒫_X is bounded with respect to any generating set.

The variant of a semigroup $S$ with respect to an element $a\in S$, denoted $S^a$, is the semigroup with underlying set $S$ and operation ⋆ defined by $x\star y=xay$ for $x,y\in S$. In this paper, we study variants ${{𝒯}}_X^a$ of the full transformation semigroup ${{𝒯}}_X$ on a finite set $X$. We explore the structure of ${{𝒯}}_X^a$ as well as its subsemigroups $\mathrm{\text{Reg}}({{𝒯}}_X^a)$ (consisting of all regular elements) and ${{ℰ}}_X^a$ (consisting of all products of idempotents), and the ideals of $\mathrm{\text{Reg}}({{𝒯}}_X^a)$. Among other results, we calculate the rank and idempotent rank (if applicable) of each semigroup, and (where possible) the number of (idempotent) generating sets of the minimal possible size.

In this paper, we study exchange rings and clean rings $R$ with $2\in \mathrm{\text{U}}(R)$ (or otherwise). Analogues of a theorem of Camillo and Yu characterizing clean and strongly clean rings with $2\in \mathrm{\text{U}}(R)$ are obtained for such rings (as well as for exchange rings) using the viewpoint of exchange equations introduced in a recent paper of the authors. We also study a new class of rings including von Neumann regular rings in which square roots of one (instead of idempotents) can be lifted modulo left ideals, and conjecture that such rings are exchange rings. This conjecture holds for commutative rings, and would hold for all rings if it holds for semiprimitive rings of characteristic $2$.

Let $R$ be a commutative ring and ${𝒜}$ an $R$-algebra. An $R$-${ℰ}$-derivation of ${𝒜}$ is an $R$-linear map of the form $I-\varphi$ for some $R$-algebra endomorphism $\varphi$ of ${𝒜}$, where $I$ denotes the identity map of ${𝒜}$. In this paper, we discuss some open problems on whether or not the image of a locally finite (LF) $R$-derivation or $R$-${ℰ}$-derivation of ${𝒜}$ is a Mathieu subspace [W. Zhao, Generalizations of the image conjecture and the Mathieu conjecture, J. Pure Appl. Algebra214 (2010) 1200–1216; Mathieu subspaces of associative algebras, J. Algebra350(2) (2012) 245–272] of ${𝒜}$, and whether or not a locally nilpotent (LN) $R$-derivation or $R$-${ℰ}$-derivation of ${𝒜}$ maps every ideal of ${𝒜}$ to a Mathieu subspace of ${𝒜}$. We propose and discuss two conjectures which state that both questions above have positive answers if the base ring $R$ is a field of characteristic zero. We give some examples to show the necessity of the conditions of the two conjectures, and discuss some positive cases known in the literature. We also show some cases of the two conjectures. In particular, both the conjectures are proved for LF or LN algebraic derivations and $R$-${ℰ}$-derivations of integral domains of characteristic zero.

In this paper we find a description, up to equivalence, of all irreducible representations for a class of algebras generated by four linearly related idempotents.

We calculate, only in terms of a commutative unital ring R of prime characteristic p and an abelian p-mixed group G, the classical Warfield q-invariants W_α,q(VR(G)) of the group VR(G) of all normalized units in the group ring R(G). This continues our results in (Extr. Math., 2005), (Collect. Math., 2008) and (J. Alg. Appl., 2008).

Let R be a commutative unital ring of arbitrary characteristic and let G be a multiplicative Abelian group. For the group ring RG we completely calculate the number (finite or infinite) of its idempotents only in terms of R, G and their sections. This strengthens our previous results in Sarajevo J. Math. (2011) and Filomat (2012).

We obtain a bound on the number of solutions of x^q = x in a finite noncommutative algebra over a field with q elements. Furthermore, we completely characterize those rings for which this maximum number is attained.

We give a description of the primitive central idempotents of the rational group algebra ℚG of a finite group G. Such a description is already investigated by Jespers, Olteanu and del Río, but some unknown scalars are involved. Our description also gives answers to their questions.

We answer multiple open questions concerning lifting of idempotents that appear in the literature. Most of the results are obtained by constructing explicit counter-examples. For instance, we provide a ring $R$ for which idempotents lift modulo the Jacobson radical $J(R)$, but idempotents do not lift modulo $J({𝕄}_2(R))$. Thus, the property “idempotents lift modulo the Jacobson radical” is not a Morita invariant. We also prove that if $I$ and $J$ are ideals of $R$ for which idempotents lift (even strongly), then it can be the case that idempotents do not lift over $I+J$. On the positive side, if $I$ and $J$ are enabling ideals in $R$, then $I+J$ is also an enabling ideal. We show that if $I⊴R$ is (weakly) enabling in $R$, then $I[t]$ is not necessarily (weakly) enabling in $R[t]$ while $I⟦t⟧$ is (weakly) enabling in $R⟦t⟧$. The latter result is a special case of a more general theorem about completions. Finally, we give examples showing that conjugate idempotents are not necessarily related by a string of perspectivities.

In [9] (which is [W. Zhao, Mathieu subspaces of associative algebras, J. Algebra350(1) (2012) 245–272]), the author takes a closer look at algebraic elements of radicals of Mathieu subspaces (of associative algebras) over a field, and suggests to look at integral elements with rings other than fields. But it seems more useful to look at so-called co-integral elements. We generalize his theory about algebraic radicals over fields to co-integral radicals over commutative rings with unity.

Furthermore, we show that over Artin rings, the concepts of integrality and co-integrality coincide. In addition, we define so-called uniform Mathieu subspaces, inspired by the fact that Mathieu subspaces with co-integral radicals are always of this type. Besides broadening the theory of [9] by means of the new concepts co-integrality and uniformity, we generalize many of the results of [9] in other ways as well. Furthermore, we obtain several new results. In the last section, we disprove a conjecture by the author of [9] (in a version of [9] prior to finding the counterexample), by showing that so-called strongly simple algebras do not need to be fields over theirselves.

We study rings in which $ab$ nonzero idempotent implies $ba$ is also an idempotent. We call such rings i-reversible. Besides studying the basic properties of i-reversible rings, we characterize i-reversible triangular matrix rings, i-reversible matrix rings over commutative rings and i-reversible exchange rings.

A clean decomposition $a=e+u$ in a ring $R$ (with idempotent $e$ and unit $u$) is said to be special if $aR\cap eR=0$. We show that this is a left-right symmetric condition. Special clean elements (with such decompositions) exist in abundance, and are generally quite accessible to computations. Besides being both clean and unit-regular, they have many remarkable properties with respect to element-wise operations in rings. Several characterizations of special clean elements are obtained in terms of exchange equations, Bott–Duffin invertibility, and unit-regular factorizations. Such characterizations lead to some interesting constructions of families of special clean elements. Decompositions that are both special clean and strongly clean are precisely spectral decompositions of the group invertible elements. The paper also introduces a natural involution structure on the set of special clean decompositions, and describes the fixed point set of this involution.

Let $R$ be a unital simple ring. Under a mild technical restriction on $R$, we characterize bilinear mappings $G:R^2\to R$ satisfying $G(u,u)u^k=u^{k}G(u,u)$, and $G(1,r)=G(r,1)=r$ for all unit $u\in R$ and $r\in R$, where $k\in \mathbb{N}$. As an application we describe bijective linear maps $𝜃:R\to R$ satisfying $𝜃(x{y}^k{x}^{-1}{y}^{-k})=𝜃(x)𝜃{(y)}^k𝜃{(x)}^{-1}𝜃{(y)}^{-k}$ for all invertible $x,y\in R$. Precisely, we will show that $𝜃$ is an isomorphism.

There has been a great deal of interest lately in studying rings in which every element can be written as a sum of potent elements and nilpotent elements. In this paper, we focus primarily on rings in which every element can be written as the sum of some number of $m$-potent elements and a nilpotent (for some choice of $m$), all of which commute with each other. The research that has been conducted thus far in this area has revealed a number of interesting results and intriguing patterns. Our goal in this paper is to unify and extend these results, for the purpose of organizing the current knowledge and facilitating future research in this area.

In this paper, we investigate zero-divisor, nilpotent, idempotent, unit, small, and irreducible elements in semiring extensions such as amount, content, and monoid semialgebras. We also introduce new concepts such as the prime avoidance property in semirings, entire-like semirings, semialgebras with Property (A), and also, Armendariz and McCoy semialgebras and we prove some results related to these concepts. For example, we prove that if $B$ is an $S$-semialgebra, then under some conditions, the set of zero-divisors $Z(B)$ of $B$ is the union of the extended maximal primes of $Z(S)$. Finally, we prove a generalization of Eisenstein’s irreducibility criterion.

Let $X$ be a finite partially ordered set, $R$ an associative unital ring and $\sigma$ an endomorphism of $R$. We describe some properties of the skew incidence ring $I(X,R,\sigma )$ such as invertible elements, idempotents, the Jacobson radical and the center. Moreover, if two skew incidence rings $I(X,R,\sigma )$ and $I(Y,S,\tau )$are isomorphic and the only idempotents of $R$ and $S$ are the trivial ones, we show that the partially ordered sets $X$ and $Y$ are isomorphic.

Let ${𝒮}$ be a finite cyclic semigroup written additively. An element $e$ of ${𝒮}$ is said to be idempotent if $e+e=e$. A sequence $T$ over ${𝒮}$ is called idempotent-sum-free provided that no idempotent of ${𝒮}$ can be represented as a sum of one or more terms from $T$. We prove that an idempotent-sum-free sequence over ${𝒮}$ of length over approximately a half of the size of ${𝒮}$ is well structured. This result generalizes the Savchev–Chen Structure Theorem for zero-sum-free sequences over finite cyclic groups.

In this paper minimal codes for several classes of non-cyclic abelian groups have been constructed by explicitly determining a complete set of primitive idempotents in the corresponding group algebras. Some classes of non-p-groups have also been considered. The minimum distances of such abelian codes have been discussed and compared to the minimum distances of cyclic codes of same lengths and dimensions over the same field.