The notion of almost Gorenstein ring given by Barucci and Fröberg [2] in the case where the local rings are analytically unramified is generalized, so that it works well also in the case where the rings are analytically ramified. As a sequel, the problem of when the endomorphism algebra m : m of m is a Gorenstein ring is solved in full generality, where m denotes the maximal ideal in a given Cohen-Macaulay local ring of dimension one. Characterizations of almost Gorenstein rings are given in connection with the principle of idealization. Examples are explored.

Abstract. The conjecture of Wolmer Vasconcelos [V] on the vanishing of the firstHilbert coefficient e 1 (Q) is solved affirmatively, where Q is a parameter ideal in a Noetherian local ring. Basic properties of the rings for which e 1 (Q) vanishes are derived.The invariance of e 1 (Q) for parameter ideals Q and its relationship to Buchsbaum rings are studied.

The set of the first Hilbert coefficients of parameter ideals relative to a moduleits Chern coefficients-over a local Noetherian ring codes for considerable information about its structure-noteworthy properties such as that of Cohen-Macaulayness, Buchsbaumness, and of having finitely generated local cohomology. The authors have previously studied the ring case. By developing a robust setting to treat these coefficients for unmixed rings and modules, the case of modules is analyzed in a more transparent manner. Another L. Ghezzi et al. series of integers arise from partial Euler characteristics and are shown to carry similar properties of the module. The technology of homological degree theory is also introduced in order to derive bounds for these two sets of numbers.

Quasi-socle ideals, that is the ideals I of the form I = Q : m q in a Noetherian local ring (A, m) with the Gorenstein tangent cone G(m) = n≥0 m n /m n+1 are explored, where q ≥ 1 is an integer and Q is a parameter ideal of A generated by monomials of a system x 1 , x 2 , · · · , x d of elements in A such that (x 1 , x 2 , · · · , x d ) is a reduction of m. The questions of when I is integral over Q and of when the graded rings G(I) = n≥0 I n /I n+1 and F(I) = n≥0 I n /mI n are Cohen-Macaulay are answered. Criteria for G(I) and R(I) = n≥0 I n to be Gorenstein rings are given.

In this paper, we explore the structure of the normal Sally modules of rank one with respect to an m-primary ideal in a Nagata reduced local ring R which is not necessary Cohen-Macaulay. As an application of this result, when the base ring is Cohen-Macaulay analytically unramified, the extremal bound on the first normal Hilbert coefficient leads to the depth of the associated graded rings G with respect to a normal filtration is at least dim R − 1 and G turns in to Cohen-Macaulay when the third normal Hilbert coefficient is vanished.2010 Mathematics Subject Classification.13A30, 13B22, 13B24, 13B30, 13D40, 13E05, 13H10.