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  • Print publication year: 2003
  • Online publication date: September 2009

5 - Principally Injective and FP Rings

Summary

A ring R is right mininjective if R-linear maps aRR extend to RRRR whenever aR is a simple right ideal. It is natural to enquire about the rings for which this condition is satisfied for all principal right ideals aR (for example if R is regular or right self-injective). These rings, called right principally injective (or right P-injective), play a central role in injectivity theory. An example is given of a right P-injective ring that is not left P-injective. If R is right P-injective it is shown that Zr = J, that R is directly finite if and only if monomorphisms RR → RR are epic, and that R has the ACC on right annihilators if and only if it is left artinian. If R is right P-injective and right Kasch then SιessRR. A semiperfect, right P-injective ring R in which Sress RR is called a right GPF ring. Hence the right PF rings are precisely the right self-injective, right GPF rings, and these right GPF rings exhibit many of the properties of the right PF rings: They admit a Nakayama permutation, they are right and left kasch, they are left finitely cogenerated, Sr = Sι is essential on both sides, and Zr = J = Zι.

Unlike mininjectivity, being right P-injective is not a Morita invariant. In fact, Mn(R) is right P-injective implies that R is right n-injective.

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