Decomposition of a ring induced by minus partial order

Abstract

Abstract. The minus partial order linear algebraic methods have proven to be useful in the study of complex matrices. This paper extends study of minus partial orders to general rings. It is shown that the condition a < − b, where < − is the minus partial order, defines two triples of orthogonal idempotents, and thus, a decomposition of a ring into a direct sum of abelian groups. Hence, several well-known results concerning minus partial order on real and complex matrices are generalized. Analogous decompositions for rectangular matrices over a ring and for Banach space operators are obtained. The equivalent conditions for the invariance of ab (1) a under the choices of b (1) are also obtained. An original inspiration for this work came from the study of minus partial order on complex matrices and from linear algebra methods.

Star, sharp, core and dual core partial order in rings with involution

“…From Theorem 2.10 it follows that e 1 ¼ ab ¼ aa ; f 1 ¼ b a ¼ a a. By Lemma 3.1 and representation for a given in(16), it follows that x 2 af1; x ij are arbitrary elements in appropriate subsets of R. Note that e 1 þ e 2 ¼ bb and f 1 þ f 2 ¼ b b. By Lemma 3.1 and representation for b given in(17), it follows that x 2 bf1; 3g if and only if…”

“…From (18) and (19), we can conclude that a and b À a have properties that is similar to invertibility of a 2 e 1 Rf 1 and b À a 2 e 2 Rf 2 . We need the following theorem which is proved in [16].…”

Partial orders in rings based on generalized inverses – unified theory

“…We now give some elementary properties of minus partial order which will be used in the sequel. If a < − b then, see [27]:…”

“…The minus and star partial orders are originally defined on a semigroup and others on the set of complex matrices. In [27] and [29] these orders are considered in the context of arbitrary ring with involution. It is shown that these orders are partial orders on appropriate subsets of R. One can notice an obvious similarity in definitions of partial orders based on generalized inverses.…”

Star, left-star, and right-star partial orders in Rickart ∗-rings

“…It is known (see [20]) that in every von Neumann regular ring a − ≤ b if and only if a = ab − a = ab − b = bb − a for some (and thus every) b − such that bb − b = b. This is true especially when for b − we take b † .…”