Research output: Contribution to journal › Article › peer-review
On a Generalization of Self-Injective Rings. / Zilberbord, I. M.; Sotnikov, S. V.
In: Vestnik St. Petersburg University: Mathematics, Vol. 53, No. 1, 01.01.2020, p. 45-51.Research output: Contribution to journal › Article › peer-review
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TY - JOUR
T1 - On a Generalization of Self-Injective Rings
AU - Zilberbord, I. M.
AU - Sotnikov, S. V.
N1 - Publisher Copyright: © 2020, Pleiades Publishing, Ltd. Copyright: Copyright 2020 Elsevier B.V., All rights reserved.
PY - 2020/1/1
Y1 - 2020/1/1
N2 - Abstract: In this work the notion of left (right) self-injective ring is generalized. We consider rings that are direct sum of injective module and semisimple module as a left (respectively, right) module above itself. We call such rings left (right) semi-injective and research their properties with the help of two-sided Peirce decomposition of the ring. The paper contains the description of left noetherian left semi-injective rings. It is proved that any such ring is a direct product of (two-sided) self-injective ring and several quotient rings (of special kind) of rings of upper triangular matrices over skew fields. From this description it follows that for left semi-injective rings we have the analogue of the classical result for self-injective rings. Namely, if a ring is left noetherian and left semi-injective then this ring is also right semi-injective and two-sided artinian.
AB - Abstract: In this work the notion of left (right) self-injective ring is generalized. We consider rings that are direct sum of injective module and semisimple module as a left (respectively, right) module above itself. We call such rings left (right) semi-injective and research their properties with the help of two-sided Peirce decomposition of the ring. The paper contains the description of left noetherian left semi-injective rings. It is proved that any such ring is a direct product of (two-sided) self-injective ring and several quotient rings (of special kind) of rings of upper triangular matrices over skew fields. From this description it follows that for left semi-injective rings we have the analogue of the classical result for self-injective rings. Namely, if a ring is left noetherian and left semi-injective then this ring is also right semi-injective and two-sided artinian.
KW - injective module
KW - Peirce decomposition
KW - self-injective ring
KW - semisimple module
UR - http://www.scopus.com/inward/record.url?scp=85082605760&partnerID=8YFLogxK
U2 - 10.1134/S106345412001015X
DO - 10.1134/S106345412001015X
M3 - Article
AN - SCOPUS:85082605760
VL - 53
SP - 45
EP - 51
JO - Vestnik St. Petersburg University: Mathematics
JF - Vestnik St. Petersburg University: Mathematics
SN - 1063-4541
IS - 1
ER -
ID: 71671848