TY - JOUR
T1 - On essential extensions of direct sums of injective modules
AU - Beidar, K. I.
AU - Ke, W. F.
N1 - Copyright:
Copyright 2017 Elsevier B.V., All rights reserved.
PY - 2002/2/1
Y1 - 2002/2/1
N2 - We characterize locally Noetherian modules MR in terms of essential extensions of direct sums of M-injective modules. As a special case (M = R) we obtain that the following conditions are equivalent: (a) R is right Noetherian; (b) any essential extension of the direct sum of any family of injective right R-modules is the direct sum of injective right R-modules; (c) any essential extension of the direct sum of the family (ℰ(Si) | i = 1, 2, ….) of injective hulls of any family (Si | i = 1, 2, ….) of simple right R-modules is the direct sum of injective right R-modules; (d) for any family (Si | i: 1, 2, ….) of simple right R-modules there exists an infinite subset ℐ of natural numbers such that ⊕i∈ℐ, ℰ(Si) is an injective module.
AB - We characterize locally Noetherian modules MR in terms of essential extensions of direct sums of M-injective modules. As a special case (M = R) we obtain that the following conditions are equivalent: (a) R is right Noetherian; (b) any essential extension of the direct sum of any family of injective right R-modules is the direct sum of injective right R-modules; (c) any essential extension of the direct sum of the family (ℰ(Si) | i = 1, 2, ….) of injective hulls of any family (Si | i = 1, 2, ….) of simple right R-modules is the direct sum of injective right R-modules; (d) for any family (Si | i: 1, 2, ….) of simple right R-modules there exists an infinite subset ℐ of natural numbers such that ⊕i∈ℐ, ℰ(Si) is an injective module.
UR - https://www.scopus.com/pages/publications/0036002657
UR - https://www.scopus.com/pages/publications/0036002657#tab=citedBy
U2 - 10.1007/s00013-002-8224-2
DO - 10.1007/s00013-002-8224-2
M3 - Article
AN - SCOPUS:0036002657
SN - 0003-889X
VL - 78
SP - 120
EP - 123
JO - Archiv der Mathematik
JF - Archiv der Mathematik
IS - 2
ER -