Intended The Even Bimodules And Generalized Matrix Ring

Annihilator plays an important role in studying dual-rings, quasi-dual rings and dual-bimodules. In chapter two we first define left quasi-dual bimodules. Assume ring R is associative with identity and module MR is a unitary right R-moaule, let S=End(MR\ it is trivial that $MR is a (S, 7?)-bimodule. A bimodule S&/R is a left quasi-dual bimodule if every essential submodule K of MR and every essential left ideal L of S satisfy withrMls(K)=K and lsrM(L)=L respectively. At first, we study the properties ofquasi-dual bimodules. If sMR is a left quasi-dual bimodule, we obtain the following conclusions:(1) rMls(Soc(MR))-Soc(MR) and lsrM(Soc(sSy)-Soc(sS);(2) If MR is a cs-module, then SOC(MR) is essential in MR,(3) If MR is non-M-singular, then MR is semisimple;(4) If MR is projective in o[M] and semisimple, then MR is non-M-singular.Next, we discuss the relations between left quasi-dual bimodules and left dual-bimodules, we obtain that a left quasi-dual bimodule is a left dual bimodule if it satisfies one of the following conditions:( i ) sM is minimal injective and MR is a M-minimal injective kasch-module;( ii ) MR is a M-minimal injective kasch-module and for any two ideals LI and L2 ofSS rM(L1 n L2)-rw(L1)+rM(I2);(iii) sM is minimal injective and for any two submodules A and B of MR,Lastly, we applicate the quasi-duality on smash product algebra R#H, and obtain an answer of the semiprime problem, i.e., let H be a finite-dimensional semisimple Hopf algebra and R be an H-module algebra, if R is left quasi-dual and semiprime, then R#H is semiprime.In chapter three we study generalized matrix rings, we first study the radical of thegeneralized matrix ring A, obtained Density Theorem and Wedderburn-Artin Theorem of generalized matrix rings. Next, we characterized some properties of directed graph by means of radical. Finally, we give the relations among the Von Neumann regular radicals of generalized matrix ring A, Aji-ring Ajjand ring A. We obtain the following conclusions.(1) If r is a super nil radical of a ring, A is a g.m.ring, then r(A) is a g.m.ideal of A.(2) If r is a super nil radical of a ring. Then r is a N-radical if and only if for anyg.m.ring.4, r(A) -(3) If A has g.m.left and right no zero divisors then g.m.rn(A) = (4) Let V = is a group graded vector space, where V& is finitely dimensional. (5) r(A) = gmr(A) = r(A) = , where r doenotes rb, rk or rL...