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Strong regularity and generalized inverses in Jordan systems. (English) Zbl 0759.17019

An associative ring $A$ is called strongly regular if $a\in A{a}^{2}$ for any $a\in A$. As shown by M. Benslimane A. Fernandez-Lopez, E. Garceia Rus and El A. M. Kaidi [Algebras Groups Geom. 6, No. 4, 353-360 (1989; Zbl 0742.17001)], strong regularity is a symmetric concept, because $A$ is strongly regular if and only if $a\in {a}^{2}A{a}^{2}$ for any $a\in A$. Therefore, this condition can be expressed in terms of the symmetrized Jordan algebra ${A}^{+}$ and extended to Jordan systems. A Jordan algebra, or triple system, $J$ is called strongly regular if $a\in P{\left(a\right)}^{2}J$ for any $a\in A$, where $P\left(a\right)$ is the usual quadratic operator.

In the paper under review, once the above remarks are carefully settled, the concept of strong regularity is investigated in the more general setting of a Jordan semigroup (a nonempty set $J$ with a map $x↦P\left(x\right)$, $P\left(x\right):J\to J$, such that $P\left(P\left(x\right)y\right)=P\left(x\right)P\left(y\right)P\left(x\right)$. An element $a$ in the Jordan semigroup $J$ is shown to be strongly regular $\left(a\in P{\left(a\right)}^{2}J\right)$ if and only if there is an element $b\in J$, called the generalized inverse of $a$, such that $a=P\left(a\right)b$, $b=P\left(b\right)a$ and $\left[P\left(a\right),P\left(b\right)\right]=0$. Then, strongly regular elements are characterized and studied in Jordan triple systems. A Jordan triple system is proved to be strongly regular if and only if it is von Neumann regular and has no nonzero nilpotent elements. The notion of generalized inverse is also shown to be closely related to the notion of the Moore-Penrose inverse in associative triple system and the group inverse in semigroups.

Finally, other related notions in associative algebras, such as abelian regularity and Drazin inverses are naturally extended and investigated in Jordan systems.

##### MSC:
 17C10 Structure theory of Jordan algebras