The purpose of this paper is to establish a real Nullstellensatz, a Positivstellensatz and a Nichtnegativstellensatz for matrices over a commutative ring. The Stellensätze in this paper may be regarded as certain generalizations of the abstract Stellensätze for commutative rings. [ABSTRACT FROM AUTHOR]
The modified quantum algebra associated to a quantum algebra was introduced by Lusztig. has a remarkable basis, which was defined by Lusztig, called the canonical basis. In this paper, we give an explicit description of all elements of the canonical basis of for type . [ABSTRACT FROM AUTHOR]
In this paper, we introduce m-fold stable ideals and show that m-fold stable range for ideals is invariant under matrix extension. Also we prove that every m-fold stable ideal is right and left symmetric. [ABSTRACT FROM AUTHOR]
In this paper, we investigate the necessary and sufficient conditions on exchange rings, under which every regular matrix admits diagonal reduction. Also we show that an exchange ring R is strongly separative if and only if for any finitely generated projective right R-module C, if A and B are any right R-modules such that 2C⊕A≅C⊕B, then C⊕A≅B. [ABSTRACT FROM AUTHOR]
We introduce in this paper a straightforward but useful method for computing indefinite rational matrix products. The method is used to prove a certain identity involving definite sums and a definite integral. [ABSTRACT FROM AUTHOR]