International Journal of Algebra and Computation

Let $G$ be a finite abelian group. As a consequence of the results of Di Vincenzo and Nardozza, we have that the generators of the $T_G$-ideal of $G$-graded identities of a $G$-graded algebra in characteristic 0 and the generators of the $T_{G\times {\mathbb{Z}}_2}$-ideal of $G\times {\mathbb{Z}}_2$-graded identities of its tensor product by the infinite-dimensional Grassmann algebra $E$ endowed with the canonical grading have pairly the same degree. In this paper, we deal with ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$-graded identities of $E_{k^{\ast }}\otimes E$ over an infinite field of characteristic $p>2$, where $E_{k^{\ast }}$ is $E$ endowed with a specific ${\mathbb{Z}}_2$-grading. We find identities of degree $p+1$ and $p+2$ while the maximal degree of a generator of the ${\mathbb{Z}}_2$-graded identities of $E_{k^{\ast }}$ is $p$ if $p>k$. Moreover, we find a basis of the ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$-graded identities of $E_{k^{\ast }}\otimes E$ and also a basis of multihomogeneous polynomials for the relatively free algebra. Finally, we compute the ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$-graded Gelfand–Kirillov (GK) dimension of $E_{k^{\ast }}\otimes E$.

The Deep Drilling Algorithm (DDA) is an efficient non-evolutionary algorithm, extracted from previous work with evolutionary algorithms, that takes as input a finite groupoid and an operation over its universe, and searches for a term representing that operation. We give theoretical and experimental evidence that this algorithm is successful for all idemprimal term continuous groupoids, which appear to be almost all finite groupoids, and that the DDA is seriously compromised or fails for most finite groupoids not meeting both of these conditions. See our online version of the DDA at http://hampshire.edu/lspector/dda.

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 $2$.

We prove that the finite representation property holds for representation by partial functions for the signature consisting of composition, intersection, domain and range and for any expansion of this signature by the antidomain, fixset, preferential union, maximum iterate and opposite operations. The proof shows that, for all these signatures, the size of base required is bounded by a double-exponential function of the size of the algebra. This establishes that representability of finite algebras is decidable for all these signatures. We also give an example of a signature for which the finite representation property fails to hold for representation by partial functions.

In this work, we realize Leavitt path algebras as partial skew groupoid rings. This yields a free path groupoid grading on Leavitt path algebras. Using this grading, we characterize free path groupoid graded isomorphisms of Leavitt path algebras that preserves generators.

We found the lower bounds on the number of ones in fully indecomposable $(0,1)$ matrices such that their permanents are equal to the determinants of matrices obtained by a suitable change of signs in the entries. We consider separately general case and the case of symmetric matrices with vanishing diagonals.

A finite group $FH$ is said to be Frobenius-like if it has a nontrivial nilpotent normal subgroup $F$ with a nontrivial complement $H$ such that $[F,h]=F$ for all nonidentity elements $h\in H$. Suppose that a finite group $G$ admits a Frobenius-like group of automorphisms $FH$ of coprime order with $[F^{\prime },H]=1.$ In case where $C_G(F)=1$ we prove that the groups $G$ and $C_G(H)$ have the same nilpotent length under certain additional assumptions.

We study some properties of the generalized local cohomology modules $H_{I,J}^i(M,N)$ of $M,N$ with respect to a pair of ideals $(I,J)$ in Serre subcategories. Some results concerning to $({𝒮},I,J)$-cofinite modules are also given in this paper.