关于环上的块循环矩阵环  

On The Ring of Block Circulant Matrices over A Ring

作　　者：贤锋[1] 

机构地区：[1]闽江学院数学系,福建福州350000

出　　处：《数学研究》2004年第3期309-313,共5页Journal of Mathematical Study

摘　　要：定义了环R上的块循环矩阵环A ,主要证明了下列结论 :(1)若J是A的理想 ,d1,d2 ,… ,dm 是R的可逆元 ,则存在R的理想I使得J =I[σ1,σ2 ,… ,σm].　　 (2 )若 d1,d2 ,… ,dm 是R的可逆元 ,则(i)R是单环当且仅当A是单环 ;(ii)R是局部环当且仅当A是局部环 ;(iii)J(A) =J(R) [σ1,σ2 ,… ,σm];(iv)R是半本原环当且仅当A是半本原环 .(3)若d1,d2 ,… ,dm 都是R的幂零元 ,则J(A) =J(R) ( (i1,i2 ,… ,im)∈T \{( 0 ,0 ,… ,0 ) }Rσi11σi22 …σimm .　　 (4 )R是左Artin(Noether)环当且仅当A是左Artin(Noether)环 .(5 )若R有左Morita对偶 (自对偶 ) ,则A有左Morita对偶 (自对偶 ) .The ring A of block circulant matrices over a ring R is defined, and the following results are proved. (1) If J is an ideal of A, and d 1,d 2,…,d m are invertible elements of R, then there is an ideal I of R such thatJ=I[σ 1, σ 2,…,σ m]. (2) If d 1,d 2,…,d m are invertible elements of R, then (i) R is a simple ring if and only if so is A; (ii) R is a local ring if and only if so is A; (iii) J(A)=J(R)[σ 1,σ 2,…,σ m]; (iv) R is a semiprimitive ring if and only if so is A. (3) If d 1,d 2,…,d m are nilpotent elements of R, thenJ(A)=J(R)I( (i 1,i 2,…,i m)∈T\｛(0,0,…,0)｝ Rσ i 1 1σ i 2 2…σ i m m). (4) R is a left Artinian (Noetherian) ring if and only if so is A. (5) If R has a left Morita duality (selfduality), then so does A.

关 键 词：块循环矩阵环 MORITA对偶 ARTIN环 NOETHER环 

分 类 号：O153.3[理学—数学]