Dimensions Of Rings And Modules, And Their Applications

This is a survey. In this theses we sommarize results about dimensions and their applications in rings and modules, The paper is separated to two parts, In the first part, we introduce dimensions in rings, especially in algebraic geometry, and in the second part the dimensions in modules is introduced, with emphasis on Goldie dimension, Krull dimension, injective dimension and finitely-generated module.First, for the dimensions in Algebraic Geometry we have:Theorem 1.1 Let Y be an affine algebraic set, Then the dimension of Y equal to the dimension of its affine coordinate ring A(Y).Theorem 1.2 Let k be a field, and B be an integral domain, which is a finitely generated k-algebra, Then(1) the dimension of B is equal to the transcend degree of quotient field k(B) to k.(2) for every prime ideals P in B, we have h(P)+dim B/P= dim B.Theorem 1.3 For a Noetherian local ring R with the msximal ideal m,the following three numbers are equal:(1) the maximum length of prime ideal chains in R;(2) the degree of eigen-polynomial Xm (n)= l(A/mn);(3) the number of least generating element of m-primary ideal.For the Krull dimension in Noether ring, we have the following:Theorem 1.4 Let R be a semi-prime ring of right Krull dimension, then(3) the cyclic left ideal is uniformly left ideal;(4) k(R)=sup {k(C) | C is the cyclic left ideal} Theorem 1.5 Let R be a prime ring of right Krull dimension, then(1)κ(R)=κ(A), for every nonzero right ideal A.(2)k(R/B)<κ(R),(?)0≠B.(3) if MR is finite, then k(M) 0, then T is a serial ring of finite Krull dimension m-1 that has the a.c.c. on right annihilators.For injective dimension and finitely-generated module, we haveProposition 1.8 For k-vector space, V the following conditions are equiva-lent:(1) finite dimension:(2) finite length:(3) a. c. c.: (4) d. c. c.:Moreovey:if these conditions are satisfied length=dimension.