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Mininjective rings. (English) Zbl 0879.16002

For a ring \(R\), a right \(R\)-module \(M\) is called mininjective if for each simple right ideal \(I\) of \(R\) every \(R\)-homomorphism \(f\colon I\to M\) extends to \(R\). A ring \(R\) is called right mininjective if \(R_R\) is mininjective; equivalently every isomorphism between simple right ideals is given by left multiplication by an element of \(R\). Mininjective rings were initially introduced by M. Harada [Osaka J. Math. 19, 587-597 (1982; Zbl 0495.16022)] and were studied for the Artinian case by M. Harada [loc.cit., and ibid. 20, 1-4 (1983; Zbl 0516.16009)]. A module is said to have squarefree socle if every nonzero homogeneous component of its socle is simple. A ring \(R\) is called right min-PF if \(R\) is semiperfect, right mininjective in which \(\text{Soc}(R_R)\) is essential in \(R_R\) and \(\ell(r(K))=K\) of every simple left ideal \(K\subseteq Re\), where \(e\) is a local idempotent.
It is shown that the right mininjectivity is a Morita invariant property, and a commutative ring is mininjective if and only if it has squarefree socle. Semiperfect right mininjective rings are completely characterized: a semiperfect ring \(R\) is right mininjective if and only if \(\text{Soc}(R_R)e\) is simple or zero for each local idempotent \(e\) of \(R\).
Furthermore several interesting results for right mininjectivity, which can extend well known results on pseudo- and quasi-Frobenius rings, are investigated. For examples: (1) If \(R\) is a semiperfect, right mininjective ring for which \(\text{Soc}(eR)\neq 0\) for every local idempotent \(e\) of \(R\), then \(R\) is right and left Kasch, and \(R\) admits a Nakayama permutation of its basic idempotents; (2) If \(R\) is a right min-PF, then \(R\) is right and left Kasch, \(J(R)=Z(R_R)\), and \(R\) admits a Nakayama permutation of its basic idempotents, where \(J(R)\) and \(Z(R_R)\) denote the Jacobson radical and the right singular ideal of \(R\), respectively. Also as an application, it is proved that a right Artinian ring \(R\) is quasi-Frobenius if and only if \(R\) is mininjective.
Reviewer: J.K.Park (Pusan)

MSC:

16D50 Injective modules, self-injective associative rings
16L60 Quasi-Frobenius rings
16L30 Noncommutative local and semilocal rings, perfect rings
16D60 Simple and semisimple modules, primitive rings and ideals in associative algebras
16P20 Artinian rings and modules (associative rings and algebras)
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