اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدUnit-regularity and representability for semiartinian *-regular rings
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تأليف
Christian Herrmann
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We show that any semiartinian *-regular ring R is unit-regular; if, in addition, R is subdirectly irreducible then it admits a representation within some inner product space.
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Let R be a unit-regular ring, and let a,b,c in R satisfy aba=aca. If ac and ba are group invertible, we prove that ac is similar to ba. Furthermore, if ac and ba are Drazin invertible, then their Drazin inverses are similar. For any ntimes n complex
matrices A,B,C with ABA = ACA ,we prove that AC and BA are similar if and only if their k-powers have the same rank. These generalize the known Flanders theorem proved by Hartwig.
Given a subdirectly irreducible *-regular ring R, we show that R is a homomorphic image of a regular *-subring of an ultraproduct of the (simple) eRe, e in the minimal ideal of R; moreover, R (with unit) is directly finite if all eRe are unit-regular
. Finally, unit-regularity is shown for every member of the variety generated by artinian *-regular rings (endowed with unit and pseudo-inversion).
We survey recent progress on the realization problem for von Neumann regular rings, which asks whether every countable conical refinement monoid can be realized as the monoid of isoclasses of finitely generated projective right $R$-modules over a von Neumann regular ring $R$.
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