By Froberg R.

ISBN-10: 0471974420

ISBN-13: 9780471974420

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**Example text**

In particular the only coclosed ideals of a principal ideal domain are the trivial ideals. We now consider further generalisations of self-projectivity. The ﬁrst of these was introduced by Brodski˘ı in [43]. 18. Intrinsically projective modules. A module M is called intrinsically projective if every diagram with exact row M Mk GN G 0, where k ∈ N and N ⊂ M , can be extended commutatively by some M → M k . 19. Characterisation of intrinsically projective modules. Let M be any module and set S = End(M ).

A coclosed submodule of a π-projective module is idempotent. It follows that any coclosed left ideal of a ring is idempotent. In particular the only coclosed ideals of a principal ideal domain are the trivial ideals. We now consider further generalisations of self-projectivity. The ﬁrst of these was introduced by Brodski˘ı in [43]. 18. Intrinsically projective modules. A module M is called intrinsically projective if every diagram with exact row M Mk GN G 0, where k ∈ N and N ⊂ M , can be extended commutatively by some M → M k .

Moreover, Ke pg ⊂ N and so pg is a small epimorphism. dim(M/N ). dim(M/N ) whenever N M. 4. Properties of hollow dimension. Let N be a submodule of M . dim(Mk ). dim(M/N ), then N M. 50 Chapter 1. dim(M ) < ∞, then any epimorphism f : M → M is small. If moreover M is semi-Hopﬁan, then f is an isomorphism. Proof. dim(Mi ). dim(M ) = ∞. dim(Mi ) = ni < ∞ for each i ≤ k. 2(e), there exist small epimorphisms fi : Mi → j=1 all 1 ≤ j ≤ ni . These yield a small epimorphism f = (f1 , . . , fk ), M f G k i=1 ni j=1 Hij G 0.

### An introduction to Grobner bases by Froberg R.

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