In 1977 Hartwig and Luh asked whether an element $a$ in a Dedekind-finite ring $R$ satisfying $aR= {a}^{2} R$ also satisfies $Ra= R{a}^{2} $. In this paper, we answer this question in the negative. We also prove that if $a$ is an element of a Dedekind-finite exchange ring $R$ and $aR= {a}^{2} R$, then $Ra= R{a}^{2} $. This gives an easier proof of Dischinger’s theorem that left strongly $\pi $-regular rings are right strongly $\pi $-regular, when it is already known that $R$ is an exchange ring.

In this paper it is shown how nonpointed exactness provides a framework which allows a simple categorical treatment of the basics of Kurosh–Amitsur radical theory in the nonpointed case. This is made possible by a new approach to semi-exactness, in the sense of the first author, using adjoint functors. This framework also reveals how categorical closure operators arise as radical theories.

We give two generalizations of Kac's Theorem on representations of quivers. One is to representations of equipped graphs by relations, in the sense of Gelfand and Ponomarev. The other is to representations of quivers in which certain of the linear maps are required to have maximal rank.

Let $(R, \mathfrak{m})$ be a Cohen–Macaulay complete local ring. We will apply an inductive argument to show that for every nonprojective locally projective maximal Cohen–Macaulay object $ \mathcal{X} $ of the morphism category of $R$ with local endomorphism ring, there exists an almost split sequence ending in $ \mathcal{X} $. Regular sequences are exploited to reduce the Krull dimension of $R$ on which the inductive argument is established. Moreover, the Auslander–Reiten translate of certain objects is described.

Let $K$ be a commutative ring with unity, $R$ an associative $K$-algebra of characteristic different from $2$ with unity element and no nonzero nil right ideal, and $f({x}_{1} , \ldots , {x}_{n} )$ a multilinear polynomial over $K$. Assume that, for all $x\in R$ and for all ${r}_{1} , \ldots , {r}_{n} \in R$ there exist integers $m= m(x, {r}_{1} , \ldots , {r}_{n} )\geq 1$ and $k= k(x, {r}_{1} , \ldots , {r}_{n} )\geq 1$ such that $\mathop{[{x}^{m} , f({r}_{1} , \ldots , {r}_{n} )] }\nolimits_{k} = 0$. We prove that: (1) if $\text{char} (R)= 0$ then $f({x}_{1} , \ldots , {x}_{n} )$ is central-valued on $R$; and (2) if $\text{char} (R)= p\gt 2$ and $f({x}_{1} , \ldots , {x}_{n} )$ is not a polynomial identity in $p\times p$ matrices of characteristic $p$, then $R$ satisfies ${s}_{n+ 2} ({x}_{1} , \ldots , {x}_{n+ 2} )$ and for any ${r}_{1} , \ldots , {r}_{n} \in R$ there exists $t= t({r}_{1} , \ldots , {r}_{n} )\geq 1$ such that ${f}^{{p}^{t} } ({r}_{1} , \ldots , {r}_{n} )\in Z(R)$, the center of $R$.

A $\ast $-ring $R$ is called (strongly) $\ast $-clean if every element of $R$ is the sum of a unit and a projection (that commute). Vaš [‘$\ast $-Clean rings; some clean and almost clean Baer $\ast $-rings and von Neumann algebras’, J. Algebra 324(12) (2010), 3388–3400] asked whether there exists a $\ast $-ring that is clean but not $\ast $-clean and whether a unit regular and $\ast $-regular ring is strongly $\ast $-clean. In this paper, we answer these two questions. We also give some characterisations related to $\ast $-regular rings.

We derive some combinatorial consequences from the positivity of Donaldson–Thomas invariants for symmetric quivers conjectured by Kontsevich and Soibelman and proved recently by Efimov. These results are used to prove the Kac conjecture for quivers having at least one loop at every vertex.

We give a negative answer to the question raised by Mart Abel about whether his proposed definition of ${K}_{0} $ and ${K}_{1} $ groups in terms of quasi multiplication is indeed equivalent to the established ones in algebraic $K$-theory.

Let $\Lambda $ be an Auslander 1-Gorenstein Artinian algebra with global dimension two. If $\Lambda $ admits a trivial maximal 1-orthogonal subcategory of $\text{mod } \Lambda $, then, for any indecomposable module $M\in \text{mod } \Lambda $, the projective dimension of $M$ is equal to one if and only if its injective dimension is also equal to one, and $M$ is injective if the projective dimension of $M$ is equal to two. In this case, we further get that $\Lambda $ is a tilted algebra.

We study the top left derived functors of the generalised $I$-adic completion and obtain equivalent properties concerning the vanishing or nonvanishing of the modules ${L}_{i} {\Lambda }_{I} (M, N)$. We also obtain some results for the sets $\text{Coass} ({L}_{i} {\Lambda }_{I} (M; N))$ and ${\text{Cosupp} }_{R} ({ H}_{i}^{I} (M; N))$.

We show that if $A$ and $H$ are Hopf algebras that have equivalent tensor categories of comodules, then one can transport what we call a free Yetter–Drinfeld resolution of the counit of $A$ to the same kind of resolution for the counit of $H$, exhibiting in this way strong links between the Hochschild homologies of $A$ and $H$. This enables us to obtain a finite free resolution of the counit of $\mathcal {B}(E)$, the Hopf algebra of the bilinear form associated with an invertible matrix $E$, generalizing an earlier construction of Collins, Härtel and Thom in the orthogonal case $E=I_n$. It follows that $\mathcal {B}(E)$ is smooth of dimension 3 and satisfies Poincaré duality. Combining this with results of Vergnioux, it also follows that when $E$ is an antisymmetric matrix, the $L^2$-Betti numbers of the associated discrete quantum group all vanish. We also use our resolution to compute the bialgebra cohomology of $\mathcal {B}(E)$in the cosemisimple case.

The stable rank of Leavitt path algebras of row-finite graphs was computed by Ara and Pardo. In this paper we extend this to an arbitrary directed graph. In part our computation proceeds as for the row-finite case, but we also use knowledge of the row-finite setting by applying the desingularising method due to Drinen and Tomforde. In particular, we characterise purely infinite simple quotients of a Leavitt path algebra.

Let $\alpha $ be any radical of associative rings. A radical $\gamma $ is called $\alpha $-like if, for every $\alpha $-semisimple ring $A$, the polynomial ring $A[x] $ is $\gamma $-semisimple. In this paper we describe properties of $\alpha $-like radicals and show how they can be used to solve some open problems in radical theory.

It follows from methods of B. Steinberg, extended to inverse categories, that finite inverse category algebras are isomorphic to their associated groupoid algebras; in particular, they are symmetric algebras with canonical symmetrizing forms.We deduce the existence of transfer maps in cohomology and Hochschild cohomology from certain inverse subcategories. This is in part motivated by the observation that, for certain categories , being a Mackey functor on is equivalent to being extendible to a suitable inverse category containing . We further show that extensions of inverse categories by abelian groups are again inverse categories.

Let be a finite category and let k be a field. We consider the category algebra and show that -mod is closed symmetric monoidal. Through comparing with a co-commutative bialgebra, we exhibit the similarities and differences between them in terms of homological properties. In particular, we give a module-theoretic approach to the multiplicative structure of the cohomology rings of small categories. As an application, we prove that the Hochschild cohomology rings of a certain type of finite category algebras are finitely generated.

We prove for a large family of rings R that their λ-pure global dimension is greater than one for each infinite regular cardinal λ. This answers in the negative a problem posed by Rosický. The derived categories of such rings then do not satisfy, for any λ, the Adams λ-representability for morphisms. Equivalently, they are examples of well-generated triangulated categories whose λ-abelianization in the sense of Neeman is not a full functor for any λ. In particular, we show that given a compactly generated triangulated category, one may not be able to find a Rosický functor among the λ-abelianization functors.

To each tagged triangulation of a surface with marked points and non-empty boundary we associate a quiver with potential in such a way that whenever we apply a flip to a tagged triangulation the Jacobian algebra of the quiver with potential (QP) associated to the resulting tagged triangulation is isomorphic to the Jacobian algebra of the QP obtained by mutating the QP of the original one. Furthermore, we show that any two tagged triangulations are related by a sequence of flips compatible with QP-mutation. We also prove that, for each of the QPs constructed, the ideal of the non-completed path algebra generated by the cyclic derivatives is admissible and the corresponding quotient is isomorphic to the Jacobian algebra. These results, which generalize some of the second author’s previous work for ideal triangulations, are then applied to prove properties of cluster monomials, like linear independence, in the cluster algebra associated to the given surface by Fomin, Shapiro and Thurston (with an arbitrary system of coefficients).

We prove a conjecture of Kontsevich, which asserts that the iterations of the non-commutative rational map Fr:(x,y)→(xyx−1,(1+yr)x−1) are given by non-commutative Laurent polynomials with non-negative integer coefficients.

We compute the center of the ring of PD differential operators on a smooth variety over ℤ/pnℤ, confirming a conjecture of Kaledin (private communication). More generally, given an associative algebra A0 over ℤp and its flat deformation An over ℤ/pn+1ℤ, we prove that under a certain non-degeneracy condition, the center of An is isomorphic to the ring of length-(n+1) Witt vectors over the center of A0.

Let $R$ be a commutative ring. The regular digraph of ideals of $R$, denoted by $\Gamma (R)$, is a digraph whose vertex set is the set of all nontrivial ideals of $R$ and, for every two distinct vertices $I$ and $J$, there is an arc from $I$ to $J$ whenever $I$ contains a nonzero divisor on $J$. In this paper, we study the connectedness of $\Gamma (R)$. We also completely characterise the diameter of this graph and determine the number of edges in $\Gamma (R)$, whenever $R$ is a finite direct product of fields. Among other things, we prove that $R$ has a finite number of ideals if and only if $\mathrm {N}_{\Gamma (R)}(I)$ is finite, for all vertices $I$ in $\Gamma (R)$, where $\mathrm {N}_{\Gamma (R)}(I)$ is the set of all adjacent vertices to $I$ in $\Gamma (R)$.