Algebra/Linear Algebra Seminar

Speaker: Furuzan Ozbek

Title: Generalized Baer's Criterion

Abstract: In 1940, Baer proved that an $R$-module $M$ is injective if and only if any homomorphism from an ideal $I$ to $M$ can be extended to a homomorphism from the ring $R$ to $M$. In relative homological algebra, we define an analogous notion to injective modules which is called a Gorenstein Injective module. It is not difficult to see that every injective module is Gorenstein injective. Our goal has been to find a similar criterion to that of Baer's for the Gorenstein injectives. Our starting point was to look at a simple case where $R$ is a local, Gorenstein ring where $dim R=1$.

Then we proved that M is Gorenstein injective if and only if $Ext^1(R/<r>,M)=0$ for all $R$-regular elements $r$. We want to generalize this result to $n$-dimensional Gorenstein rings.