Decompositions of matrices into potent and square-zero matrices

Abstract

In order to find a suitable expression of an arbitrary square matrix over an arbitrary finite commutative ring, we prove that every such matrix is always representable as a sum of a potent matrix and a nilpotent matrix of order at most two when the Jacobson radical of the ring has zero-square. This somewhat extends results of ours in Linear Multilinear Algebra (2022) established for matrices considered on arbitrary fields. Our main theorem also improves on recent results due to Abyzov et al. in Mat. Zametki (2017), Šter in Linear Algebra Appl. (2018) and Shitov in Indag. Math. (2019).

Rings close to periodic with applications to matrix, endomorphism and group rings

“…only remains to show is that such a decomposition can be preserved in the whole ring M n (R), but the procedure of lifting potent elements and even tripotents is rather more complicated than the standard trick for lifting idempotents (compare with the methods described in [26] and [43]). At this stage, we are unable to do that, so that this should be treated as a left-open problem.…”

Decomposition of matrices into invertible and square-zero matrices

“…Nevertheless, such a decomposition does not hold for fields of zero characteristic; in fact, take for example A = 2I, where I is the identity matrix -if it could be written as P + N with N 2 = 0, then P = 2I − N clearly satisfies the relation (P − 2I) 2 = 0 and, on the other hand, it satisfies the relation P k − P = 0, because P is potent -but then the minimal polynomial of P must divide these two polynomials which is definitely not possible. Further insight in that matter over some special finite rings was achieved by us in [9]. We also refer the interested reader to [8] for some other aspects of the realization of matrices into the sum of specific elements over certain fields.…”

“…Our motivating tool in writing up this article is to continue our investigations in [10] and [9] by refining the obtained there matrix's decompositions and, concretely, to explore when any square matrix over an arbitrary finite or infinite field is presentable as a sum of an invertible matrix (in particular, of a torsion unit matrix) and a square-zero matrix, thus somewhat improving the aforementioned result from [7] too. So, to achieve this purpose of ours, we manage the further work into two subsequent sections.…”