Ring Of Linear Transformations Research Articles

A Galois-Type Correspondence Theory for Actions of Finite-Dimensional Pointed Hopf Algebras on Prime Algebras

The objective of this paper is to present a Galois-type correspondence theory for X-outer actions of finite-dimensional Hopf algebras over a field k on prime algebras R. Noncommutative Galois theory was initiated by Noether in her work on inner automorphisms of simple algebras and was w x further studied by Jacobson J1 , who established a Galois correspondence w theory for a finite group of automorphisms acting on a division ring. In J2, x Chap 6 Jacobson used the product of the ring and the group of automorphisms to develop a Galois theory for a ring of linear transformations of a vector space. A major advance is due to Kharchenko, who developed a Galois correspondence theory for actions of automorphisms and derivaŽ . tions on prime semiprime rings. His definitions of inner and outer actions, which were named X-outer and X-inner actions, involve the Martindale ring of quotients of a prime ring R. Montgomery and Passman presented the X-outer case separately and used trace forms of minimal length to show in an elegant way Kharchenko’s Galois correspondence

Quasi-critical ring of a primitive ring with nonzero socle

IN ref. [1] Jacobson proved the structure theorem for primitive rings with nonzero socles that R is a primitive ring withsocle S≠{0} if and only if there is a pair of dual vector spaces (M,M’) over a division ring Δ such that S=F(M, M’)(?) R(?)(?)(M, M’), where (?)(M, M’)-{ω∈Ω|ωM’(?)M’, Ω is the complete ring of linear transformations of M over Δ}, F(M, M’) is the set of all linear transformations of (?)(M, M’) of finite rank. After that, some people reproved this theorem by using different methods such as those in refs. [2, 3]. In this

Isomorphisms of semigroups of transformations

If M is a centered operand over a semigroup S, the suboperands of M containing zero are characterized in terms of S‐homomorphisms of M. Some properties of centered operands over a semigroup with zero are studied.A Δ‐centralizer C of a set M and the semigroup S(C, Δ) of transformations of M over C are introduced, where Δ is a subset of M. When Δ = M, M is a faithful and irreducible centered operand over S(C, Δ). Theorems concerning the isomorphisms of semigroups of transformations of sets Mi over Δi‐centralizers Ci, i = 1, 2 are obtained, and the following theorem in ring theory is deduced: Let Li, i = 1, 2 be the rings of linear transformations of vector spaces (Mi, Di) not necessarily finite dimensional. Then f is an isomorphism of L1 → L2 if and only if there exists a 1 − 1 semilinear transformation h of M1 onto M2 such that fT = hTh−1 for all T ∈ L1.

On a finite dimensional quasi-simple module

1. If R is a ring and M is a (right) A'-module such that MR^O, then M is said to be quasi-simple provided that (Ql) the endomorphism (module endomorphism) ring of the quasiinjective hull of M is a division ring, (Q2) if N is a nonzero submodule of M then there is a nonzero endomorphism / of M such that /(M) C N. Any simple module is clearly quasi-simple, however a quasi-simple module is not necessarily simple. For example, if R is a semiprime ring with a uniform right ideal U such that the right singular ideal of R is zero then the regular A'-module U is quasi-simple since the endomorphism ring of its quasi-injective hull is a division ring [5, 1.7, p. 263] and for any nonzero submodule N oi U there is xEN such that O^acf/CJV. Let us say that a quasi-simple module is finite dimensional if its quasi-injective hull is a finite dimensional vector space over its endomorphism ring. The main results in this paper are the following: Let ¥ be a faithful quasi-simple A'-module for some ring R and let M be the quasi-injective hull of M. Let D = Hom/j(iif, M) and if N is a nonzero submodule of M then define K(N) = Ylomn(N, N). Then for any nonzero submodule N of M, K(N) is a right order in D, M=DN and if {7wt-}=1 is a finite sequence of Z?-linearly independent elements in M and if [y}=1 is a finite sequence in M, then there exist rER, 0^&GHomfi(Af, M) such that 0^k\NEK(N) and mtr = kyi for all l^if$n. If, in addition, the singular submodule of M is zero then, for any large right ideal B of R one can choose rG-B and 0^i£HomR(M, M) such that 0^k\N EK(N) and mir = kyt for all l^ijSw. In case the set {mi, m2, • • • , mn is a basis for M then r is a regular element. If R is a right Goldie prime ring and U is a uniform right ideal then U is a finite dimensional quasi-simple module. Conversely, if A* has a faithful finite dimensional quasi-simple right module M then A3 is a right Goldie prime ring, and M is isomorphic to a uniform right ideal of R. Hence our theory above provides a new proof for Theorem 10 of [3, p. 603]. Let Dn be the nXn matrix ring representing r\omD(M, M) relative to the basis {mi, m2, • ■ ■ , mn] and K(N)n be the nXn matrix ring over K(N). Let R* be the subring of Dn which represents R as a ring of linear transformations on M over D relative to the same basis.

Some remarks on 𝜈-transitive rings and linear compactness

Johnson in [3] has introduced the concept of a v-transitive ring which generalizes the notion of a dense ring of linear transformations. We give necessary and sufficient conditions that an abstract ring be isomorphic to a v-transitive ring which contains finite-valued linear transformations. The condition (2) used here is a modification of one used by Baer [I ] in his characterization of the endomorphism ring of a primary Abelian operator group. This condition is also related to the linear compactness of a ring considered as a right module over itself. This enables one to conclude that a primitive ring with minimal ideals which is linearly compact (in any topology in which it is a topological ring) is the ring of all linear transformations of a vector space, and that a primitive Banach algebra is linearly compact only when it is finite dimensional. A ring E(F, A) of linear transformations of the vector space A over the division ring F is called i-transitive if to every set of less than K, elements aj of A, linearly independent over F, and any set of elements bj of A, in one-one correspondence with the aj, there exists a transformation oin E(F, A), such that ajo = bj for all j. Let K be an abstract ring and P an arbitrary subset thereof. The right ideal of all elements k in K which satisfies Pk = 0 shall be called a right annulet. Now let W= W(K) be the class of all right annulets which are cross-cuts of a finite number of maximal right annulets of K. By a W-coset is meant a coset of an ideal in the set W(K). If E(F, A) is any ring of linear transformations and S is a subspace of A, we denote by R(S) the totality of transformations oCE(F, A) satisfying So=0.

On the Theory of Primitive Rings

In a recent paper2 we have called a ring 2 primitive if 2 contains a maximal right ideal a such that the quotient (a: W) = (0). The quotient (: 1) is defined to be the largest two-sided ideal of 21 that satisfies the condition 21(a: 21) C a. Thus if 21lias an identity, (a: 21) is the largest two-sided ideal of 21 contained in a. Primitive rings appear to play a role in the general structure theory of rings analogous to that played by simple rings in the Wedderburn-Artin theory. Primitive rings are also fundamental in general representation theory since it is known that a necessary and sufficient condition that a ring 21 be primitive is that 21 be isomorphic to an irreducible ring W of endomorphisms in a commutative group T. If 2 is irreducible in T and Zi is the division ring of endomorphisms commutative with the elements of A, then R is a dense ring of linear transformations in T regarded as a vector space over Z.3 We shall call 2 a left primitive ring if 2 contains a maximal left ideal a' such that the quotient (}': 21)l = (0). Here (s': 2) 1 is the largest two-sided ideal in 2 having the property (a': 21)1 21 C a'. Evidently, a ring is left primitive if and only if it is anti-isomorphic to an irreducible ring of endomorphisms. It is an open question whether or not a primitive ring is necessarily left primitive. This is indeed the case if 2 contains minimal ideals. In many other respects the theory of primitive rings that contain minimal ideals constitutes the most satisfactory part of the theory of primitive rings. An appropriate tool for studying rings of this type is a generalization of the duality theory previously used by Dieudonn6 in studying simple rings that possess minimal ideals.4 In this note we develop this generalization of Dieudonn6's results. We also investigate certain natural topologies that can be defined in any primitive ring.