Let X be a magma, that is a set equipped with a binary operation, and consider a function α: X → X. We say that X is Hom-associative if, for all x, y, z ∈ X, the equality α(x)(yz) = (xy)α(z) holds. For every isomorphism class of magmas of order two, we determine all functions α making X Hom-associative. Furthermore, we find all such α that are endomorphisms of X. We also consider versions of these results where the binary operation on X and the function α only are partially defined. We use our findings to construct numerous examples of two-dimensional Hom-associative as well as multiplicative magma algebras.  

We generalize a classical result by Passman on primeness of unital strongly group graded rings to the class of nearly epsilon-strongly group graded rings which are not necessarily unital. Using this result, we obtain (i) a characterization of prime s-unital strongly group graded rings, and, in particular, of infinite matrix rings and of group rings over s-unital rings, thereby generalizing a well-known result by Connell; (ii) characterizations of prime s-unital partial skew group rings and of prime unital partial crossed products; (iii) a generalization of the well-known characterizations of prime Leavitt path algebras, by Larki and by Abrams-Bell-Rangaswamy.  

We obtain criteria for when a ring with enough idempotents is left/right artinian or noetherian in terms of local criteria defined by the associated complete set of idempotents for the ring. We apply these criteria to object unital category graded rings in general and, in particular, to the class of skew category algebras. Thereby, we generalize results by Nastasescu-van Oystaeyen, Bell, Park, and Zelmanov from the group graded case to groupoid, and in some cases category, gradings. 

We investigate properties of group gradings on matrix rings 𝑀𝑛⁡(𝑅), where R is an associative unital ring and n is a positive integer. More precisely, we introduce very good gradings and show that any very good grading on 𝑀𝑛⁡(𝑅) is necessarily epsilon-strong. We also identify a condition that is sufficient to guarantee that 𝑀𝑛⁡(𝑅) is an epsilon-crossed product, i.e. isomorphic to a crossed product associated with a unital twisted partial action. In the case where R has IBN, we provide a characterization of when 𝑀𝑛⁡(𝑅) is an epsilon-crossed product. Our results are illustrated by several examples.

The results that are stated in P. Nystedt and J. Öinert [Group gradations on Leavitt path algebras, J. Algebra Appl. 19(9) (2020) 2050165, Sec. 4] hold true, but due to an oversimplification some of the proofs are incomplete. The purpose of this note is to amend and complete the affected proofs.

We present a streamlined, slightly modified version, in the two-variable situation, of a beautiful, but not so well known, theory by Bögel [1, 2], already from the 1930s, on an alternative higher dimensional calculus of real functions, a double calculus, which includes many two-variable extensions of classical results from single variable calculus, such as Rolle’s theorem, Lagrange’s mean value theorem, Cauchy’s mean value theorem, Fermat’s extremum theorem, the first derivative test, and the first and second fundamental theorems of calculus. 

The results that are stated in P. Nystedt and J. Oinert [Group gradations on Leavitt path algebras, J. Algebra Appl. 19(9) (2020) 2050165, Sec. 4] hold true, but due to an oversimplification some of the proofs are incomplete. The purpose of this note is to amend and complete the affected proofs.

We study Ore extensions of non-unital associative rings. We provide a characterization of simple non-unital differential polynomial rings R[x; delta], under the hy-pothesis that R is s-unital and ker(delta) contains a non-zero idempotent. This result gener-alizes a result by oinert, Richter and Silvestrov from the unital setting. We also present a family of examples of simple non-unital differential polynomial rings.

Suppose that R is an associative unital ring and that E= (E-0, E-1, r, s) is a directed graph. Using results from graded ring theory, we show that the associated Leavitt path algebra L-R(E) is simple if and only if R is simple, E-0 has no nontrivial hereditary and saturated subset, and every cycle in E has an exit. We also give a complete description of the centre of a simple Leavitt path algebra.