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In 2012, Ananthnarayan, Avramov and Moore give a new construction of Gorenstein rings from two Gorenstein local rings, called their connected sum. Given a Gorenstein ring, one would like to know whether it decomposes as a connected sum and if so, what are its components. We answer these questions in the Artinian case and investigate conditions on the ring which force it to be indecomposable as a connected sum. We further give a characterization for Gorenstein Artin local rings to be decomposable as connected sums, and as a consequence, obtain results about its Poincare series and minimal number of generators of its defining ideal. Finally, in the graded case, we show that the indecomposable components appearing in the connected sum decomposition are unique up to isomorphism.
A new construction of rings is introduced, studied, and applied. Given surjective homomorphisms $Rto Tgets S$ of local rings, and ideals in $R$ and $S$ that are isomorphic to some $T$-module $V$, the emph{connected sum} $R#_TS$ is defined to be the l
In 2012, Ananthnarayan, Avramov and Moore gave a new construction of Gorenstein rings from two Gorenstein local rings, called their connected sum. In this article, we investigate conditions on the associated graded ring of a Gorenstein Artin local ri
A connected sum construction for local rings was introduced in a paper by H. Ananthnarayan, L. Avramov, and W.F. Moore. In the graded Artinian Gorenstein case, this can be viewed as an algebraic analogue of the topological construction of the same na
It is proved that the minimal free resolution of a module M over a Gorenstein local ring R is eventually periodic if, and only if, the class of M is torsion in a certain Z[t,t^{-1}]-module associated to R. This module, denoted J(R), is the free Z[t,t
A result of Foxby states that if there exists a complex with finite depth, finite flat dimension, and finite injective dimension over a local ring $R$, then $R$ is Gorenstein. In this paper we investigate some homological dimensions involving a semid