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We study affine semigroup rings as algebras over subsemigroup rings. From this relative viewpoint with respect to a given subsemigroup ring, the fibered sum of two affine semigroup algebras is constructed. Such a construction is compared to the tensor product and to the classical gluings of affine semigroup rings as defined in Rosales (1997). While fibered sum can always be achieved, gluings of affine semigroup rings do not always exist. Therefore, we further investigate when the fibered sum of affine semigroup algebras gives rise to a gluing. A criterion is recovered in terms of the defining semigroups under which the gluing may take place.

Over a regular local ring of dimension two with maximal ideal m, we study the Buchsbaum-Rim multiplicity of a finitely generated module M of finite colength in a free module F. First, we investigate the colength of an m-primary ideal and its Hilbert-Samuel multiplicity using linkage theory. As applications, we establish several multiplicity formula that express the Buchsbaum-Rim multiplicity of M in terms of the Hilbert-Samuel multiplicities of ideals related to a certain minimal reduction U of M. Moreover, there exists m-primary Bourbaki ideals I and J of the modules F and M respectively such that F/M is isomorphic to I/J. We also have a formula for the Buchsbaum-Rim multiplicity whenever such I and J are given. The latter part is related to a graphical interpretation of the multiplicities in the case of monomial ideals., Comment: submitted for publication