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On \(p\)-injective rings. (English) Zbl 0847.16005

An \(R\)-module \(M_R\) is called a \(p\)- (resp., \(fp\)-) injective module if for any principal right ideal \(X\) of \(R\) (resp. any finitely generated submodule \(Y\) of \(R^{(A)}_R\)) every homomorphism \(X_R\to M_R\) (resp. \(Y_R\to M_R\)) can be extended to a homomorphism from \(R_R\) to \(M_R\) (resp. from \(R^{(A)}_R\) to \(M_R\)). A ring \(R\) is right \(p\)- (resp. \(fp\)-) injective if \(R_R\) is \(p\)- (resp. \(fp\)-) injective.
The authors show, among other results, that (i) for any serial ring \(R\), right \(p\)-injectivity and right \(pf\)-injectivity are equivalent; (ii) any semiperfect right duo right \(p\)-injective ring is right continuous; (iii) if every factor ring of a ring \(R\) is right \(p\)-injective (briefly, a right \(cp\)-injective ring), then the lattice of two-sided ideals of \(R\) is distributive; (iv) a right duo ring \(R\) is a direct sum of (two-sided) uniserial rings with nil Jacobson radical iff \(R\) is right \(cp\)-injective with no infinite set of orthogonal idempotents.

MSC:

16D50 Injective modules, self-injective associative rings
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