Sets of Elements that Pairwise Generate a Matrix Ring

Abstract

In this article, we investigate the relationship between the minimum number of proper subgroups of GL(n, q) whose union is the whole GL(n, q) and the maximum number of elements that pairwise generate GL(n, q). We show that the minimum number of proper subrings of M_n(q) whose union is the whole M_n(q) is exactly the maximum number of elements that pairwise generate M_n(q)

Maximal Subrings and Covering Numbers of Finite Semisimple Rings

“…Interestingly, the work done in [17,Sec. 7] and [5] relied on a classification of the maximal subrings of Mn(Fq) that did not include Type III maximal subrings (see [17,Lem. 7.1], which is equivalent to Racine's classifications [21,22] of maximal Fq-subalgebras of Mn(Fq)).…”

“…7.1], which is equivalent to Racine's classifications [21,22] of maximal Fq-subalgebras of Mn(Fq)). However, the techniques employed in [17] and [5] were robust enough that Type III maximal subrings where automatically excluded from the minimal covers that were formed, and so σ(Mn(Fq)) can be determined by using only Type I and Type II maximal subrings.…”

“…Less attention has been paid to covering numbers for rings. All rings with covering number three were determined in [18], [27] gives formulas for covering numbers of direct products of finite fields, and a formula for σ(Mn(Fq)) was determined in [17] and [5] (see Theorem 5.2 of the present paper). The survey paper [13] gives an overview of theorems related to coverings for various algebraic structures, including groups and rings.…”

The covering numbers of rings

“…Indeed, Kappe [18, p. 87] further writes, "An interesting question would be if there are integers n > 2 that are not the covering number of a ring." Other recent works on this problem include [5,9,27,32,34].…”

“…The classification of commutative unital σ-elementary rings was done in [32], and the covering numbers of the rings in parts (1) and (2) were determined in [34]. The formula for the covering number of M n (q) is due to Crestani, Lucchini, and Maróti [9,23]. Since M n (q) is simple, it is clearly σ-elementary, and the fact that these are the only noncommutative semisimple σ-elementary rings follows from the classification of covering numbers for finite semisimple rings done in [27].…”