M. P. Drazin

Lattice properties of the ∗-order for complex matrices

Abstract It is shown that the set C m × n of complex m × n matrices forms a lower semilattice under the partial ordering A ⩽ B defined by A ∗ A = A ∗ B, ∗ AA ∗ = BA ∗ , where A ∗ denotes the conjugate transpose of A. As a special case of a result for division rings, it is further shown that, over any field F, for m = n = 2 and any proper involution ∗ of F2 × 2, the corresponding intersections A ∩ B all exist.

Commuting properties of generalized inverses

Abstract For any semigroup and any , in 2012, the author defined to be a -inverse of if , and . For given , any such is unique if it exists, and, by choosing and appropriately, one can arrange for to become, inter alia, either the author’s pseudo-inverse or (if is a -semigroup) the Moore–Penrose inverse . For any , the author proved in 1958 that implies whenever exists. In this article it is shown that, more generally, implies whenever and both exist. Still more generally, a corresponding result is proved for -inverses; in particular, for the Moore–Penrose inverse, and together imply . For any , new connections between the generalized invertibility of and are also obtained.

A partial order in completely regular semigroups

is a partial order on S (qua set) iff S is weakly separative. Moreover, weakly separative semigroups occur quite widely: for example, every semigroup regular in the sense of von Neumann is weakly separative, as is every semigroup with a “proper involution” and every semiprime ring. Another widely applicable natural partial order (denoted here by JV) has recently been introduced by Nambooripad [ 123 and independently by Hartwig [ 111, who define aNb for given a, b E S iff there is some x E S such that

Rings with Central Idempotent or Nilpotent Elements

It is easy to see ( cf . Theorem 1 below) that the centrality of all the nilpotent elements of a given associative ring implies the centrality of every idempotent element; and (Theorem 7) these two properties are in fact equivalent in any regular ring. We establish in this note various conditions, some necessary and some sufficient, for the centrality of nilpotent or idempotent elements in the wider class of π-regular rings (in Theorems 1, 2, 3 and 4 the rings in question are not even required to be π-regular).

GENERALIZATIONS OF FITTING'S LEMMA IN ARBITRARY ASSOCIATIVE RINGS

While Fitting’s lemma has been stated in varying degrees of generality, all versions known to the writer deal with arbitrary automorphisms a of a suitably restricted module V (i.e. a 2 EndV ). We note first that, if one replaces EndV by an arbitrary associative ring R, then one can quite easily express the essence of Fitting’s lemma as asserting a property of elements a of R. Moreover, while previous versions of Fitting’s lemma require that V satisfy some finiteness restriction(s), and while one could easily imitate this approach in the more general R-context, one may obtain more precise results by letting R be any (non-commutative) associative ring and by asking what conditions a given element a 2 R must satisfy in order for this property to hold for a; in other words, one may consider ‘‘local Fitting properties’’ of individual elements a 2 R rather than (or as well as) global properties of R (or V ) as a whole. In this spirit the main contributions of this article are results which, for certain elements a of arbitrary associative rings R, decompose a as the sum a 1⁄4 c þ n (e.g. uniquely for all a 2 R whenever R is artinian or commutative noetherian) of orthogonal summands c; n 2 R (i.e. with cn 1⁄4 nc 1⁄4 0) having strongly dissimilar properties generalizing (in a precise sense) those in Fitting’s lemma: differently put, c comes close to being invertible, while n comes close to being nilpotent (‘‘closeness’’ in each

Commutators in associative rings

Let R be an arbitrary associative ring, and X a set of generators of R . The elements of X generate a Lie ring, [ X ], say, with respect to the addition and subtraction in R , and the multiplication [ a, b ] = ab − ba . In this note we shall be concerned with the following question: if [ X ] is given to be nilpotent as a Lie ring, what does this imply about R ?

The Nilpotence of Nil Subrings

1. It is well kinown that, in any associative ring R with minimal condition oii (say) left ideals, every nil left ideal is nilpotent (for the simplest lknown proof of tllis, due to R. Brauer, see [4, Theorem 13, p. 64]) ; more recently, Jacobson has shown ([5], and cf. also [4, Theorem 30, p. 71]) that every nil subring of R, besides also certain types of (nil) subsets not admitting the additioin alnd mnultiplication of R, must be nilpotent. We recall also [4, Theorem 29, p. 71] that, at least in the presence of a two-sided identity element, the minimal condition on left ideals implies the maximal condition oni these ideals. Our main result in this note is that, in any associative ring with maximal condition on nilpotent subrings, every nil subring must in fact be nilpotent. Indeed, we establish the stronger resuilt stated as Theorem 1 below; this may be compared with a result of Levitzki [6], who showed that, if R has maximal conditioni on both left and right ideals, then every nil subring of R is nilpotent (and that every nil ideal is nilpotent under a weaker, though more complicated, hypothesis; cf. also [7]). In a concluding section, we establisl-h some analogous results for Lie rings. We note first a trivial lemma, which serves as a starting-point for our arguments in both the succeeding sections:

Faithfully ⋆-representing semigroups and groupoids with involution by infinite matrices

The set M ∞(Z) of countably infinite square matrices over Z forms a (partial, non-associative) groupoid. The authors, with Davies, showed in 1996 that every finite or countable groupoid embeds in M ∞(Z). This is a non-associative analogue of the elementary result that every finite or countable semigroup embeds in row-finite matrices over Z, although the proof is very different. This article considers corresponding questions about ⋆-embeddings of groupoids (G, ⋆) with involution ⋆. The two main results here are that (i) there is an improper involution ⊛ on M ∞(Z) such that every countable (G, ⋆) embeds in (M ∞(Z), ⊛), and (ii) every countable partial ⋆-groupoid obtained by omitting all products g ⋆ g from (G, ⋆) embeds in (M ∞(Z), T), where T is the transpose. It is also noted that, even in the finite associative case, not every finite semigroup with proper involution ⋆-embeds in an M n (Z) with a proper involution. However, with respect to a version of ⊛ acting on square matrices of finite even order, as an analogue of (i) apparently new even for the finite associative case, it is shown that (iii) every finite ⋆-semigroup embeds into some (M 2n (Z), ⊛). All these results have corresponding versions for k- and (k, ⋆)-algebras. As an application, these embedding results are used to provide a much simpler and more general way of finding examples, like those constructed by Sivakumar in 2006, of infinite matrices having uncountably many distinct groups and Moore–Penrose generalized inverses which are not classical inverses.

Subclasses of (b, c)-inverses

For any semigroup S and any , the author [Linear Algebra Appl. 2012;436:1909–1923] introduced the idea of the (b, c)-inverse y of a, and then [Bicommuting properties of generalized inverses. Linear Multilinear Algebra, to appear] of the subclass consisting of all ‘standard’ y. The present article discusses four further subclasses, each still including the generalized inverses y respectively introduced by Bott–Duffin, Moore–Penrose and the author. All the inclusions between these classes are determined, and examples are given to prove all the non-inclusions.

Diagonalizability of traces

Abstract For any finite-dimensional algebra A over a field F it is shown that the bilinear forms associated with the characters of the left and right regular representations of A (indeed, of all representations of A ) can be simultaneously diagonalized.