Rothโs equivalence problem in unit regular rings
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- by Robert E. Hartwig PDF
- Proc. Amer. Math. Soc. 59 (1976), 39-44 Request permission
Abstract:
It is shown that for matrices over a unit regular ring $\left [ {\begin {array}{*{20}{c}} A & C \\ 0 & D \\ \end {array} } \right ]\sim \left [ {\begin {array}{*{20}{c}} A & 0 \\ 0 & D \\ \end {array} } \right ]$ if and only if there exist solutions $X$ and $Y$ to $AX - YD = C$, thus providing a partial generalization to Rothโs theorem.References
- Adi Ben-Israel and Thomas N. E. Greville, Generalized inverses: theory and applications, Pure and Applied Mathematics, Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1974. MR 0396607
- Gertrude Ehrlich, Unit-regular rings, Portugal. Math. 27 (1968), 209โ212. MR 266962
- Gertrude Ehrlich, Units and one-sided units in regular rings, Trans. Amer. Math. Soc. 216 (1976), 81โ90. MR 387340, DOI 10.1090/S0002-9947-1976-0387340-0 H. Flanders and H. Wimmer, รber die Matrizen gleichungen $AX - XB = C$ and $AX - YB = C$, Arch. Math. (to appear).
- L. Fuchs, On a substitution property of modules, Monatsh. Math. 75 (1971), 198โ204. MR 296096, DOI 10.1007/BF01299099
- F. R. Gantmacher, Matrizenrechnung. II. Spezielle Fragen und Anwendungen, Hochschulbรผcher fรผr Mathematik, Band 37, VEB Deutscher Verlag der Wissenschaften, Berlin, 1959 (German). MR 0107647
- Robert E. Hartwig, Block generalized inverses, Arch. Rational Mech. Anal. 61 (1976), no.ย 3, 197โ251. MR 399124, DOI 10.1007/BF00281485 โ, Generalized inverses, $EP$ elements and associates (to appear).
- Robert E. Hartwig and Jiang Luh, On finite regular rings, Pacific J. Math. 69 (1977), no.ย 1, 73โ95. MR 437586, DOI 10.2140/pjm.1977.69.73
- Melvin Henriksen, On a class of regular rings that are elementary divisor rings, Arch. Math. (Basel) 24 (1973), 133โ141. MR 379574, DOI 10.1007/BF01228189
- Nathan Jacobson, Lectures in abstract algebra. Vol. II. Linear algebra, D. Van Nostrand Co., Inc., Toronto-New York-London, 1953. MR 0053905
- Carl D. Meyer Jr., Generalized inverses of block triangular matrices, SIAM J. Appl. Math. 19 (1970), 741โ750. MR 272795, DOI 10.1137/0119075
- Marvin Rosenblum, The operator equation $BX-XA=Q$ with self-adjoint $A$ and $B$, Proc. Amer. Math. Soc. 20 (1969), 115โ120. MR 233214, DOI 10.1090/S0002-9939-1969-0233214-7
- William E. Roth, The equations $AX-YB=C$ and $AX-XB=C$ in matrices, Proc. Amer. Math. Soc. 3 (1952), 392โ396. MR 47598, DOI 10.1090/S0002-9939-1952-0047598-3
- Ivan Vidav, Modules over regular rings, Math. Balkanica 1 (1971), 287โ292. MR 289565
Additional Information
- © Copyright 1976 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 59 (1976), 39-44
- MSC: Primary 16A30; Secondary 15A21
- DOI: https://doi.org/10.1090/S0002-9939-1976-0409543-4
- MathSciNet review: 0409543