Polynomial Algebra And Its Applications

B. Buchberger introduced the concepts of term orders in the set of all power products and presented an algorithm for finding a special generators, a reduced Grobner basis which is unique for a given term order, of a given ideal of a polynomial ring over a field from any generators of the ideal by means of S-polynomials. So many problems on ideals can be solved by the theory of the Grobner basis. Since 1980s, many mathematicians have been engaged in studying the applications of the Grobner basis such as solving the system of algebraic equations, factoring polynomials, testing primary ideals, factoring algebraic manifolds, decoding circular codes in corrected codes and algebraically geometrical codes, analyzing and synthesizing high dimensional linear recurring arrays in cryptology, dealing with multidimensional systematic theory, signaling, solving integer programming and so on.The theory of Morita duality comes from the theory of the dual space of a linear space, which was proposed by K. Morita and G. Azumaya in 1950s. Since then, the theory of Morita duality has become one of the important fields of Ring Theory and Module Theory. Many algebraists are engaged in studying this theory. They study the duality and selfduality of rings such as extensions of a ring with a Morita duality(or selfduality), Noether rings, series rings, endomorphism rings and so on. C. Menini and A. D. Rio introduced a graded Morita duality into Graded Ring Theory and obtained the similar characterizations of the Morita duality.The aim of this paper is to study the applications of Grobner bases in finding the minimal polynomial of a given matrix and its inverse if it is nonsingular and to discuss selfdualities over a polynomial ring. The main results are listed in the following: It is proved that the ring consisting of all block circulant matrices over a field is isomorphic to a factor ring of a polynomial ring in multivariables over the same field. So finding the minimal polynomial of a block circulant matrix is transformed into computing the reduced Grobner basis of a kernel of a ring homomorphism. Hence an algorithm for the minimal polynomial of a block circulant matrix over the field is presented. A sufficient and necessary condition for determining singularity of a block circulant matrix over a field is given, and an algorithm for the inverse of a nonsingular block circulant matrix over the field is presented. A sufficient and necessary condition for determining singularity of a block circulant matrix over a quaternion division algebra over a field is given, and twoalgorithms for the inverse of a nonsingular block circulant matrix over the quaternion division algebra are presented. An algorithm for Grobner basis for an ideal of a polynomial ring over a group algebra of a finite group over a field is given. An integral algebraic linear programming is defined and an algorithm for solving the programming is given. Algorithms for the minimal polynomial and the inverse of a given block circulant matrix over a group algebra of a finite group over a field are presented, and a method of determining singularity of this block circulant matrix is given. Algorithms for the minimal polynomial and the inverse of a given block symmetric circulant matrix over a group algebra of a finite group over a field are presented. and a method of determining singularity of this block symmetric circulant matrix is given. The definition of a mixed block matrix is given, and an algorithm for the inverse of a given mixed block matrix over a group algebra of a finite group over a field is presented, and a method of determining singularity of this mixed block matrix is given. Graded triangular extensions and graded trivial extensions over a ring are defined respectively, and the relation between them with grade Morita dualities and their initial subrings with Morita dualities is discussed. The graded selfduality of a polynomial ring in multivariables over a cogenerator ring as a graded ring of type different group i...