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.