Related papers: Going Up and Lying Over in Congruence--modular Alg…
We show that under certain conditions, well-studied algebraic properties transfer from the class $\mathcal{Q}_{_\text{RFSI}}$ of the relatively finitely subdirectly irreducible members of a quasivariety $\mathcal{Q}$ to the whole…
We introduce a generic expression language describing behaviours of finite coalgebras over sets; besides relational systems, this covers, e.g., weighted, probabilistic, and neighbourhood-based system types. We prove a generic Kleene-type…
Tilting modules, generalising the notion of progenerator, furnish equivalences between pieces of module categories. This paper is dedicated to study how much these pieces say about the whole category. We will survey the existing results in…
We introduce and study certain hyperbolic versions of automorphic Lie algebras related to the modular group. Let $\Gamma$ be a finite index subgroup of $\mathrm{SL}(2,\mathbb{Z})$ with an action on a complex simple Lie algebra $\mathfrak…
The literature in persistent homology often refers to a "structure theorem for finitely generated graded modules over a graded principal ideal domain". We clarify the nature of this structure theorem in this context.
In this note, we study the arithmetic nature of values of modular functions, meromorphic modular forms and meromorphic quasi-modular forms with respect to arbitrary congruence subgroups, that have algebraic Fourier coefficients. This…
For a finite-dimensional representation V of a group G we introduce and study the notion of a Lie element in the group algebra k[G]. The set L(V) \subset k[G] of Lie elements is a Lie algebra and a G-module acting on the original…
Primary decomposition of commutative monoid congruences is insensitive to certain features of primary decomposition in commutative rings. These features are captured by the more refined theory of mesoprimary decomposition of congruences,…
We obtain a characterization of left perfect rings via superstability of the class of flat left modules with pure embeddings. $\mathbf{Theorem.}$ For a ring $R$ the following are equivalent. - $R$ is left perfect. - The class of flat left…
The theory of partition congruences has been a fascinating and difficult subject for over a century now. In attempting to prove a given congruence family, multiple possible complications include the genus of the underlying modular curve,…
We classify all tilting classes over an arbitrary commutative ring via certain sequences of Thomason subsets of the spectrum, generalizing the classification for noetherian commutative rings by…
We prove that Ramanujan-type congruences for integral weight modular forms away from the level and the congruence prime are equivalent to specific congruences for Hecke eigenvalues. In particular, we show that Ramanujan-type congruences are…
Modules for sesquiads and congruence schemes are introduced. It is shown that the corresponding categories are belian and that base change functors establish an ascent datum which allows for a cohomology theory to be established.
General coherence theorems are constructed that yield explicit presentations of categorical and algebraic objects. The categorical structures involved are finitary discrete Lawvere 2-theories, though they are approached within the language…
This is a companion article to my papers on Kazhdan-Lusztig polynomials and character formulae for the Lie superalgebras gl(m|n) (much revised!) and q(n). The goal is to develop the general theory of tilting modules for Lie superalgebras,…
In this book i treat linear algebra over division ring. A system of linear equations over a division ring has properties similar to properties of a system of linear equations over a field. However, noncommutativity of a product creates a…
Two observations in support of the thesis that trusses are inherent in ring theory are made. First, it is shown that every equivalence class of a congruence relation on a ring or, equivalently, any element of the quotient of a ring $R$ by…
Let R be a commutative ring with unity, M be an unitary R-module and {\Gamma} be a simple graph. This research article is an interplay of combinatorial and algebraic properties of M . We show a combinatorial object completely determines an…
The geometric and algebraic theory of monomial ideals and multigraded modules is initiated over real-exponent polynomial rings and, more generally, monoid algebras for real polyhedral cones. The main results include the generalization of…
We study relationships between certain algebraic properties of groups and rings definable in a first order structure or $*$-closed in a compact $G$-space. As a consequence, we obtain a few structural results about $\omega$-categorical rings…