Carlsen [‘$\ast $-isomorphism of Leavitt path algebras over $\Bbb Z$’, Adv. Math. 324 (2018), 326–335] showed that any $\ast $-homomorphism between Leavitt path algebras over $\mathbb Z$ is automatically diagonal preserving and hence induces an isomorphism of boundary path groupoids. His result works over conjugation-closed subrings of $\mathbb C$ enjoying certain properties. In this paper, we characterise the rings considered by Carlsen as precisely those rings for which every $\ast $-homomorphism of algebras of Hausdorff ample groupoids is automatically diagonal preserving. Moreover, the more general groupoid result has a simpler proof.

Twisted étale groupoid algebras have recently been studied in the algebraic setting by several authors in connection with an abstract theory of Cartan pairs of rings. In this paper we show that extensions of ample groupoids correspond in a precise manner to extensions of Boolean inverse semigroups. In particular, discrete twists over ample groupoids correspond to certain abelian extensions of Boolean inverse semigroups, and we show that they are classified by Lausch’s second cohomology group of an inverse semigroup. The cohomology group structure corresponds to the Baer sum operation on twists.

We also define a novel notion of inverse semigroup crossed product, generalizing skew inverse semigroup rings, and prove that twisted Steinberg algebras of Hausdorff ample groupoids are instances of inverse semigroup crossed products. The cocycle defining the crossed product is the same cocycle that classifies the twist in Lausch cohomology.

We prove that $A_{R}(G)\otimes _{R}A_{R}(H)\cong A_{R}(G\times H)$ if $G$ and $H$ are Hausdorff ample groupoids. As part of the proof, we give a new universal property of Steinberg algebras. We then consider the isomorphism problem for tensor products of Leavitt algebras, and show that no diagonal-preserving isomorphism exists between $L_{2,R}\otimes L_{3,R}$ and $L_{2,R}\otimes L_{2,R}$. In fact, there are no unexpected diagonal-preserving isomorphisms between tensor products of finitely many Leavitt algebras. We give an easy proof that every $\ast$-isomorphism of Steinberg algebras over the integers preserves the diagonal, and it follows that $L_{2,\mathbb{Z}}\otimes L_{3,\mathbb{Z}}\not \cong L_{2,\mathbb{Z}}\otimes L_{2,\mathbb{Z}}$ (as $\ast$-rings).

We realize Leavitt ultragraph path algebras as partial skew group rings. Using this realization we characterize artinian ultragraph path algebras and give simplicity criteria for these algebras.

We study the injective hulls of faithful characteristic zero finite dimensional irreducible representations of uniform nilpotent pro-p groups, seen as modules over their corresponding Iwasawa algebras. Using this we prove that the kernels of these representations are classically localisable.

Dans plusieurs articles, A. R. Prince développe une représentation d’un plan projectif fini par un anneau commutatif unitaire dont les propriétés algébriques dépendent de la structure géométrique du plan. Dans un autre article, il étend cette représentation aux designs symétriques. Cependant D.-S. Yin fait remarquer que la multiplication définie dans ce cas ne peut être associative que si le design est un plan projectif. Dans cet article on mènera une étude de cette représentation dans le cas des designs symétriques. On y montrera comment on peut faire associer un anneau commutatif unitaire à tout design symétrique; on y précisera certaines de ses propriétés, en particulier, celles qui relèvent de son invariance. On caractérisera aussi les géométries projectives finies de dimension supérieure moyennant cette représentation.

We investigate the behavior of the quasi-Baer and the right FI-extending right ring hulls under various ring extensions including group ring extensions, full and triangular matrix ring extensions, and infinite matrix ring extensions. As a consequence, we show that for semiprime rings $R$ and $S$, if $R$ and $S$ are Morita equivalent, then so are the quasi-Baer right ring hulls ${{\widehat{\text{Q}}}_{\mathfrak{q}}}\mathfrak{B}(R)$ and ${{\widehat{\text{Q}}}_{\mathfrak{q}}}\mathfrak{B}(S)$ of $R$ and $S$, respectively. As an application, we prove that if unital ${{C}^{*}}$-algebras $A$ and $B$ are Morita equivalent as rings, then the bounded central closure of $A$ and that of B are strongly Morita equivalent as ${{C}^{*}}$-algebras. Our results show that the quasi-Baer property is always preserved by infinite matrix rings, unlike the Baer property. Moreover, we give an affirmative answer to an open question of Goel and Jain for the commutative group ring $A[G]$ of a torsion-free Abelian group $G$ over a commutative semiprime quasi-continuous ring $A$. Examples that illustrate and delimit the results of this paper are provided.

For a fixed parabolic subalgebra 𝔭 of we prove that the centre of the principal block 𝒪0𝔭 of the parabolic category 𝒪 is naturally isomorphic to the cohomology ring H*(ℬ𝔭) of the corresponding Springer fibre. We give a diagrammatic description of 𝒪0𝔭 for maximal parabolic 𝔭 and give an explicit isomorphism to Braden’s description of the category PervB(G(k,n)) of Schubert-constructible perverse sheaves on Grassmannians. As a consequence Khovanov’s algebra ℋn is realised as the endomorphism ring of some object from PervB(G(n,n)) which corresponds under localisation and the Riemann–Hilbert correspondence to a full projective–injective module in the corresponding category 𝒪0𝔭. From there one can deduce that Khovanov’s tangle invariants are obtained from the more general functorial invariants in [C. Stroppel, Categorification of the Temperley Lieb category, tangles, and cobordisms via projective functors, Duke Math. J. 126(3) (2005), 547–596] by restriction.

An element $a$ in a ring $R$ with identity is called strongly clean if it is the sum of an idempotent and a unit that commute. And $a\in R$ is called strongly $\pi$-regular if both chains $aR\supseteq a^2R\supseteq\cdots$ and $Ra\supseteq Ra^2\supseteq\cdots$ terminate. A ring $R$ is called strongly clean (respectively, strongly $\pi$-regular) if every element of $R$ is strongly clean (respectively, strongly $\pi$-regular). Strongly $\pi$-regular elements of a ring are all strongly clean. Let $\sigma$ be an endomorphism of $R$. It is proved that for $\varSigma r_ix^i\in R[[x,\sigma]]$, if $r_0$ or $1-r_0$ is strongly $\pi$-regular in $R$, then $\varSigma r_ix^i$ is strongly clean in $R[[x,\sigma]]$. In particular, if $R$ is strongly $\pi$-regular, then $R[[x,\sigma]]$ is strongly clean. It is also proved that if $R$ is a strongly $\pi$-regular ring, then $R[x,\sigma]/(x^n)$ is strongly clean for all $n\ge1$ and that the group ring of a locally finite group over a strongly regular or commutative strongly $\pi$-regular ring is strongly clean.

A ring $R$ with identity is called strongly clean if every element of $R$ is the sum of an idempotent and a unit that commute. For a commutative local ring $R$, $n=3,4$, and $m, k, s \in {\mathbb N}$ it is proved that ${\mathbb M}_n(R)$ is strongly clean if and only if ${\mathbb M}_n(R[[x]])$ is strongly clean if and only if ${\mathbb M}_n(R[[x_1, x_2, \ldots, x_m]])$ is strongly clean if and only if $ {\mathbb M}_n(\frac{R[x]}{(x^{k})})$ is strongly clean if and only if $ {\mathbb M}_n(\dfrac{R[x_{1}, x_{2}, \ldots , x_{s}]}{(x^{n_1}_{1}, x^{n_{2}}_{2}, \ldots , x^{n_{s}}_{s})}) $ is strongly clean if and only if ${\mathbb M}_n(R \propto R)$ is strongly clean where $ R\propto R=\{\scriptsize(\begin{array}{@{}c@{\quad}c@{}} a& b \\ 0& a \end{array}): a, b \in R \}$ is the trivial extension of $R$. This extends a result of J. Chen, X. Yang and Y. Zhou [$\mathbf{5}$] from $n=2$ to 3 and 4.

Let $G$ be a compact $p$-adic analytic group and let $\Lambda_G$ be its completed group algebra with coefficient ring the $p$-adic integers $\mathbb{Z}_p$. We show that the augmentation ideal in $\Lambda_G$ of a closed normal subgroup $H$ of $G$ is localisable if and only if $H$ is finite-by-nilpotent, answering a question of Sujatha. The localisations are shown to be Auslander-regular rings with Krull and global dimensions equal to dim $H$. It is also shown that the minimal prime ideals and the prime radical of the $\mathbb{F}_p$-version $\Omega_G$ of $\Lambda_G$ are controlled by $\Omega_{\Delta^+}$, where $\Delta^+$ is the largest finite normal subgroup of $G$. Finally, we prove a conjecture of Ardakov and Brown [1].

Let $R$ be a ring with 1 and $P(R)$ the periodic radical of $R$. We obtain necessary and sufficient conditions for $P(\text{RG})=0$ when $\text{FG}$ is the group ring of an $\text{FC}$ group $G$ and $R$ is commutative. We also obtain a complete description of $P\left( I(X,R) \right)$ when $I(X,R)$ is the incidence algebra of a locally finite partially ordered set $X$ and $R$ is commutative.

This paper represents a general study of the (Yoneda) Ext-algebra A* of a finite dimensional K-algebra A. Our motivation lies in the problem of establishing conditions under which (i) the species of A* coincides with the dual species of A and (ii) the quasi-heredity of A (or A*) yields the quasi-heredity of A* (or A, respectively). These questions are closely related to the Kazhdan—Lusztig Theory as presented by [CPS2]. The main results include introducing the concept of a solid algebra and the relevant Theorem 4.5 as well as a rather complete description of the situation in the case of monomial algebras in Section 5.

Quasi-hereditary algebras were introduced by Cline-Parshall-Scott (see [CPS] or [PS]) to deal with highest weight categories which occur in the study of semi-simple complex Lie algebras and algebraic groups. In fact, the quasi-hereditary algebras which appear in these applications enjoy a number of additional properties. The objective of this brief note is to describe a class of lean quasi-hereditary algebras [ADL] which possess such typical characteristics. A study of these questions originated in collaboration with C. M. Ringel (see [DR]).

Necessary and sufficient conditions are given for a generalised matrix ring to be semiprime right Goldie.