arXiv subjects

Dominik Krasula

Publications and source records attributed to Dominik Krasula.

6 recordsLinked to original sources

arXiv2025

Semiperfect rings with a Nakayama permutation: A survey of Double annihilator property and Size condition

For a semiperfect ring with essential socles, the Double annihilator property encodes that the top and socle have anti-isomorphic lattices of submodules, whereas the Size condition encodes that they are isomorphic as modules. Interest in both concepts, particularly for finite rings, was revived by coding theory, where they characterise QF rings and Frobenius rings, respectively. However, their shared origins date back to the work of T. Nakayama. We study these concepts through the lens of the Nakayama permutation, an invariant initially used to define (quasi-)Frobenius rings. We propose semiperfect rings as the setting for this study, treating them as the natural generalisation of finite rings, because they possess the characteristic decomposition of unity preserved by projection onto a semisimple top. This allows us to extend the utility of the Nakayama permutation beyond the classical Artinian setting. By analysing the Nakayama permutation in this broader context, we show that many classical properties of (quasi-)Frobenius rings are not exclusive to the finite case, but are special cases of the general behaviour of semiperfect rings with essential socles. We illustrate these results using B. J. M\"uller's representation of semiperfect rings as rings of formal matrices. The clear description of socles and tops in this setting provides a straightforward method for constructing counterexamples, such as quasi-Frobenius rings that are not Frobenius.

math.RA↗

arXiv2025

Formal matrix representations of pseudo-Frobenius and Frobenius rings

Rings with Nakayama permutations, pseudo-Frobenius and Frobenius rings in particular, are studied by applying the general theory of formal matrix rings to their Peirce decompositions. A combinatorial criterion is given to decide whether a formal matrix ring with local rings on the diagonal has a prescribed Nakayama permutation. It is shown that a pseudo-Frobenius ring R can be represented as a block matrix ring, where the blocks on the diagonal are pseudo-Frobenius rings corresponding to cycles in the Nakayama permutation of R. All possible supports of such blocks are characterised. In the finite case, their local corner rings are shown to be isomorphic. We characterise local corners of quasi-Frobenius rings as a subclass of rings with a Morita self-duality. The duality contexts between these corners then appear on the shifted diagonal of their formal matrix representations. Using the combinatorial criterion, we give, under mild assumptions, a method of how to glue two rings with a Nakayama permutation. It is then used to construct an indecomposable Frobenius ring with two simple modules whose rings of endomorphisms are not isomorphic.

math.RA↗

arXiv2024

Endomorphism rings of simple modules and block decomposition

A left and right noetherian semiperfect ring R is known to be indecomposable if and only if its factor by the second power of Jacobson radical is. This characterisation is used to study simple R-modules in terms of their Ext groups. It is shown that if R is indecomposable, all its simple modules are either finite or have the same infinite cardinality and their endomorphism rings have the same characteristics. The results are further strengthened in the case when R is quasi-Frobenius.

math.RA↗

arXiv2024

M\"obius function for modules and thin representations

This paper studies the M\"obius function and related questions about the finiteness of the poset of submodules of semisimple and general modules. We show how to calculate the M\"obius function for semisimple modules based on endomorphism rings of simple submodules. We discuss the M\"obius function for representations of bounded path algebras in detail.

math.RA↗

arXiv2024

Generalised Gabriel-Roiter measure and thin representations

For Dynkin and Euclidean quivers, it is shown that Gabriel-Roiter measures of thin representations equal the induced chain length functions on the corresponding system of subquivers. This allows a combinatorial procedure to find a GR filtration of thin representations, showing that GR measures of thin representations are field-independent. It is proved that an indecomposable filtration of a thin representation is a GR filtration for a suitable choice of a length function on the category of finite-dimensional representations.

math.RT↗

arXiv2022

Restricted minimum condition in reduced commutative rings

We say that a commutative ring R satisfies the restricted minimum (RM) condition if for all essential ideals I in R, factor R/I is an Artinian ring. We will focus on Noetherian reduced rings because in this setting known results for RM domains generalize well. However, as we will show, RM rings need not be Noetherian and may have nilpotent elements. One of the classic results in the theory of RM rings is that for Noetherian domains RM condition corresponds to having Krull dimension at most one. We will show that this can be generalized to reduced Noetherian rings, thus proving that affine rings corresponding to curves are RM. We will give examples showing that the assumption that the ring is reduced is not superfluous. We will prove that CDR domains are RM and this will allow us to give a new characterization of Dedekind domains. Examples of RM rings for various classes of rings will be given. In particular, we will show that a ring of polynomials R[x] is RM if and only if R is reduced Artinian ring. And we will study the relation between RM rings and UFDs.

math.AC↗