Given a summand a(n), we seek the "indefinite sum" S(n) determined (within an additive constant) by [Formula: see text] or, equivalently, by [Formula: see text] An algorithm is exhibited which, given a(n), finds those S(n) with the property [Formula: see text] With this algorithm, we can determine, for example, the three identities [Formula: see text] [Formula: see text] and [Formula: see text]...

Let R be a commutative ring, (f) an ideal of R, and E = K(f ;R) the Koszul complex. We investigate the structure of the Tate construction T associated with E. In particular, we study the relationship between the homology of T , the quasi-complete intersection property of ideals, and the complete intersection property of (local) rings.

Robinson introduced a quotient space of pairs unbounded convex sets which share their recession cone. In this paper minimal sets, i.e., representations elements Robinson's spaces, are investigated. The fact that pair having property translation is reduced proved. the case two-dimensional formula for finding an equivalent given, criterion minimality presented, reducibility all proved, and nontri...

Let K be an algebraically closed field of characteristic zero and let I = (f1, . . . , fn) be a homogeneous R+-primary ideal in R := K[X, Y, Z]. If the corresponding syzygy bundle Syz(f1, . . . , fn) on the projective plane is semistable, we show that the Artinian algebra R/I has the Weak Lefschetz property if and only if the syzygy bundle has a special generic splitting type. As a corollary we...

A module M is said to belifting if, for anysubmodule N of M, there exists a direct summand X contained in such that N/X small M/X. satisfy the finite internal exchange propertyif, any and sum decomposition M=Lni=1Mi, M′i Mi (i= 1,2, . , n) M=X⊕(Lni=1M′i). In this paper, we first give characterizations forthe square hollow uniform be lifting (extending). addition, solve negatively question "Does...

‎In 1970‎, ‎Menegazzo [Gruppi nei quali ogni sottogruppo e intersezione di sottogruppi massimali‎, ‎ Atti Accad‎. ‎Naz‎. ‎Lincei Rend‎. ‎Cl‎. ‎Sci‎. ‎Fis‎. ‎Mat‎. ‎Natur. 48 (1970)‎, ‎559--562.] gave a complete description of the structure of soluble $IM$-groups‎, ‎i.e.‎, ‎groups in which every subgroup can be obtained as intersection of maximal subgroups‎. ‎A group $G$ is said to have the $FM$...