Abstract.W. V. Vasconcelos and M. Auslander have studied the homological properties of coherent (noetherian) rings. In [3] some theorems for regular local rings were generalized to noetherian semilocal rings. The main aim of this paper is to discuss coherent semilocal rings. HOMOLOGICAL DIMENSIONS OF COHERENT SEMILOCAL RINGSLet R be a commutative ring with identity element, and let J be the Jacobson radical of R. The concepts and the notations that are used in this paper are consistent with those in [14].Definition. Let A be an 7<-module. A normal .4-sequence is an ordered sequence ux, u2, ... , un in J such that zz, is not a zero divisor in A and, for j > 1, each ui is not a zero divisor in A/(ux, ... , u¡_x)A. We write codR(A) -n if there exists a normal ^-sequence with n terms but no normal sequence with more than n terms. Theorem 1.1. Let R be a coherent semilocal ring such that J is finitely generated, and let A be a finitely presented R-module. Then

Abstract.It is shown that a commutative coherent domain cannot have Ng dimension 2 and a commutative coherent regular ring of Ng dimension 2 cannot have finite indecomposable decomposition.

Abstract.Let R be a commutative indecomposable coherent ring. Then the following statements are equivalent: (i) R is a GCD domain; (ii) RM is a GCD domain for every maximal ideal of M of R , and every finitely generated projective ideal in R is principal; (iii) every two-generated ideal in R has finite projective dimension, and every finitely generated projective ideal in R is principal. Auslander-Buchsbaum's Theorem, etc. can be obtained from the result above.Let Ä bea commutative ring with 1 ^ 0 in this paper. R is called a GCD domain if every two elements a, b € R have a greatest common divisor (denoted by [a, b]) in R. R is said to be indecomposable if 1 is the only nonzero idempotent of R. A coherent ring is a ring whose finitely generated ideals are finitely presented.Definition. We say that R has PPC if every finitely generated projective ideal in R is a principal ideal.Theorem. Let R be an indecomposable coherent ring. Then the following statements are equivalent:(i) R is a GCD domain.(ii) Rm is a GCD domain for every maximal ideal M of R, and R has PPC.(iii) Every two-generated ideal in R has finite projective dimension, and R has PPC. Proof of Theorem, (i) =s> (ii) Let M be a maximal ideal of R. Let f , * e R, and write c = [a, b]. if y is a common divisor of y and * in Rm then d\sa

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Abstract. In this paper, we will generalize the result in weighted integrability to include all positive non-integers γ connecting with derivatives of the sum-functions.Mathematics subject classification (2010): 42A25, 42A50.