Generalized Rings And Applications To Multiplicative Semigroups And Adjoint Semigroups

A ring has naturally several structures, such as the additive group, the multiplicativesemigroup, the unit group, the adjoint semigroup, the adjoint group. It is important to describethese structures and the relationship between rings and these structures. Generally, there aretwo aspects:(1) to investigate the auxiliary structures of particular rings;(2) to investigate rings with special auxiliary structures.The additive group and the unit group of a ring have been broadly investigated. Theadjoint group of a ring is also studied by many people. However, the research on the semigroupstructures of a ring is touched on a few aspects. The goal of this thesis is to deal with themultiplicative semigroup and the adjoint semigroup of a ring in terms of generalized rings to usetechniques of universal algebra.The multiplicative semigroup of a ring is of evident importance, and the adjoint semigroupplays also an important role in a ring. In 1942, Perlis introduced the notion of quasi-regularelements. For an element x, if there exists y such that x + xy + y = x + yx + y = 0, then x iscalled quasi-regular. In 1945, Jacobson extended the idea to general rings and introduced circlemultiplication defined as follows.It is clear that (R,O) is a monoid with identity 0. (R,O), abbreviated, R?, is called the circlesemigroup or the adjoint semigroup of R. Many people also define the circle multiplication asa O b = a + b + ab. In fact, the two definitions are essentially the same up to an isomorphism.Jacobson has proved that R is a Jacobson radical ring if and only if R? is a group. Since thencircle multiplication play an significant role in ring theory.For a ring with 1, its adjoint semigroup and multiplicative semigroup are isomorphic. Ingeneral, the adjoint semigroup of a ring can be isomorphic to a subsemigroup of the multiplicative semigroup of a ring with 1. However, this point of view is not too helpful to investigate theadjoint semigroup. Therefore, it is necessary to study the adjoint semigroup directly.The circle multiplication of a ring satisfies the generalized distributive laws:So, general system (R,+,·) satisfying the generalized distributive laws have been discussed byseveral people in di?erent terms, such as (m,n)-districtive ring, pseudo ring, weak ring, quasi-ring, prering and so on.This thesis investigates mainly the multiplicative semigroup and the adjoint semigroup ofa ring in terms of universal algebra by introducing a common generalization of them.Introduction include a brief survey on multiplicative semigroups, unit groups, adjoint semi-groups, circle groups and generalization of adjoint multiplication.In Chapter 1, we give the notion of generalized ring to unify the study of multiplicativesemigroups and adjoint semigroups and use tools of universal algebra. The lattice of congruencesof generalized rings is proved to be isomorphic to a sublattice of ideals of a ring, and generalizedrings with idempotents are characterized by means of generalized matrix rings.Definition 1.1.3 Let (R,Ï„,·) be a universal algebra with a ternary operationÏ„and abinary operation"·". If (R,Ï„,·) satisfies the following conditions:(1) (R,Ï„) is a ternary group;(2) (R,·) is a semigroup;(3) aÏ„(b,c,d) =Ï„(ab,ac,ad),Ï„(b,c,d)a =Ï„(ba,ca,da), for any a,b,c,d∈R,then (R,Ï„,·) is called a generalized ring.Proposition 1.1.1 Let (A,Ï„) be a ternary group. Fix an element 0 in A. Define a + b =Ï„(a,0,b) for any a,b∈A. Then (A,+) is an Abelian group such that 0 is identity. Moreover,the Abelian groups obtained by di?erent choice of 0 are isomorphic to each other. Furthermore,Ï„(a,b,c) = a ? b + c, for a,b,c∈A.Theorem 1.1.1 Let (R,Ï„,·) be a generalized ring, and fix an element 0∈R. DefineThen (R,+,?) is an associative ring(called a ring of generalized ring R), denoted by (R,+,?)or simply R, and any two rings of (R,Ï„,·) are isomorphic to each other. Furthermore,"·"is ageneralized adjoint multiplication of the ring.Let Con(R) denote the lattice of congruences of the generalized ring R, and C(R) denotethe lattice of compatible ideals of the ring R. Theorem 1.2.1 The lattices Con(R) and C(R) are isomorphic. Particularly, Con(R) is amodular lattice.Theorem 1.2.2 Let R be a generalized ring, and I be an ideal of R. Then the followingstatements are equivalent.(1) R has an ideal A such that I = A ? A;(2) I is a compatible di?erence ideal;(3) I is a compatible ideal such that factor algebra R/σis a generalized ring with zero,whereσis the congruence correspondent to I.Proposition 1.2.5 A generalized ring R is simple if and only if R is a simple ring.Theorem 1.3.1 A generalized ring R has idempotents if and only if a ring of R is the2×2 generalized matrix ring, and R =～R11.Corollary 1.3.1 A generalized ring R has central idempotents if and only if R =～R1×R0,where R1 is a generalized ring with identity, and R0 is a generalized ring with zero.Theorem 1.3.2 Let R = M(S,T,U,V ), R = M11(S,T,U,V ), and I be an ideal of R.Then I is a compatible ideal of R if and only if I = S1 U1V1 T1 , where(1) S1 is an ideal of S, T1 is an ideal of T, U1 is an S-T submodule of U, V1 is a T-Ssubmodule of V ;(2) U1V,UV1 ? S1, and V1U,V U1 ? T1;(3) S1U,UT1 ? U1, and V S1,T1V ? V1.In Chapter 2, we introduce the notion of generalized module, give some properties ofgeneralized module, characterize the lattice of congruences of generalized modules, and discussthe chain conditions for congruences of generalized modules.Proposition 2.1.1 Let M be a generalized module over R, and M be an additive groupof M. Definewhere 0R and 0M denote the zeros of R and M, respectively. Then M is a right R-module,called a module of M.Theorem 2.1.1 Let R be a generalized ring, (M,Ï„) be an R-generalized module. DefineΛ(x) = x0R ? 0M0R, P(a) = 0Ma ? 0M0R, E = 0M0R, for any x∈M, a∈R.Then(1)Λis an endomorphism of the additive group (M,+), and P is a homomorphism fromthe additive group of R to the additive group (M,+); Let Con(M) denote the lattice of congruences of a generalized module M, and C(M) denotethe lattice of compatible submodules of M.Theorem 2.2.1 The lattices C(M) and Con(M) are isomorphic. Therefore, Con(M) is amodular lattice.Proposition 2.2.3 Let M be an R-generalized module, and K,N are generalized submod-ules of M. Then the following statements hold.(1) is a compatible submodule of M, and K?K = x?K, for x∈K.Furthermore, K ? K is isomorphic to a module of K.(2) Particularly, then K = N if and only ifProposition 2.2.4 Let M be a module of M, and N be a submodule of M. Then thefollowing statements are equivalent.(1) N = I ? I for some generalized submodule I of M;(2) N is a compatible submodule and there is a∈M such that a ? ar∈N, for any r∈R.Theorem 2.3.1 Let M be an R-generalized module. Then M satisfies DCC(respectively,ACC) for generalized submodules if and only if M satisfies DCC(respectively, ACC) for di?erencesubmodules.Theorem 2.3.2 If M satifies DCC(respectively, ACC) for principal di?erence submodules,then M satisfies DCC(respectively, ACC) for principal generalized submodules, and converselyif M does not contain infinite sets consisting of disjoint generalized submodules.Theorem 2.3.3 M satisfies DCC(respectively, ACC) for congruences is and only if M isan Artinian (respectively, Noetherian) module.Lemma 2.3.4 Let R be a generalized ring, andσbe a congruence of R. If R satisfiesDCC for principal right congruences, then so does R/σ.Lemma 2.3.5 Let R be a generalized ring. If R satisfies DCC for principal right congru-ences, and R is torsion-free, then R is divisible and right s-unital. Furthermore, R satisfies DCCfor principal right ideals.Lemma 2.3.6 If R is a generalized ring satisfying DCC for principal right congruences, then R = T⊕D2, where T is the torsion ideal of R and D is the divisible ideal of R.Lemma 2.3.7 Let R be a generalized ring. If R satisfies DCC for principal right congru-ences, and R is torsion, then R satisfies DCC for principal right ideals.Theorem 2.3.4 Let R be a generalized ring. Then R satisfies DCC for principal rightcongruences if and only if R satisfies DCC for principal right ideals.Chapter 3 is devoted to the study of regularity, complete regularity, commuting regularityand periodicity of generalized rings, and adjoint regularity of rings which is a sum of two subrings.Let M(Ω,Ω₁,U ,V ) be a Morita context ring, where ?,?1 are rings with 1. Let R and Sbe generalized subrings of ? and ?1, respectively, U and V be additive subgroups of U and V ,respectively, satisfyingLet R = S UV T . Then R is a generalized subring of M(?,?1,U ,V ).Theorem 3.1.1 Let R = S UV T as above. Then R is regular if and only if S,Tare both regular, and for any idempotents e∈S, f∈T and any u∈U, v∈V , there existRij be an n×n generalized matrix ring. Then R is regularif and only if for any i,j and for any aij∈Rij there exist bji∈Rji such that aij = aijbjiaij.Corollary 3.1.2 Let R be a ring. If there exists an idempotent e in R such thatare regular in R, then R is regular.Corollary 3.1.3(Neumman) A ring R is regular if and only if the matrix ring Mn×n(R)is regular.Corollary 3.1.4 A generalized ring R is regular if and only if R =～M11(S,T,U,V ),where S is adjoint regular, T is regular, and for any idempotents e∈S, f∈T and for any such that Corollary 3.1.5(Finogenova) If R is adjoint regular ring, then Mn(R) is adjoint regular.Theorem 3.2.1 Let R = M(S,T,U,V ) be a Morita context ring. Then the multiplicativesemigroup R of E11-generalized ring of R is a union of groups if and only if S is a generalizedradical ring, T is a strongly regular ring, V U = 0, and eU = V e = 0 for any idempotent e∈S.Theorem 3.2.2 The multiplicative semigroup of a generalized ring R is a union of groupsif and only if R～= M11(S,T,U,V ), where S is a generalized radical ring, T is a strongly regularring, V U = 0, and eU = V e = 0 for any idempotent e∈S.Corollary 3.2.1 A ring R has a generalized adjoint semigroup which is a union of groupsif and only if R is a generalized radical ring.Theorem 3.3.3 Let R be a generalized ring. Then R is commuting regular if and only ifthe multiplicative semigroup of R is isomorphic towhere S is a ring whose adjoint semigroup is a Cli?ord semigroup, U is a strongly regular ring,and N is a zero ring.Theorem 3.3.4 Let R be a generalized ring. If R is a torsion and periodic ring, then Ris periodic.Theorem 3.3.5 Let R be a periodic generalized ring. If (R,+) is torsion-free, then R isa direct product of a left zero generalized ring and a right zero generalized ring.Theorem 3.4.1 Let J be the Jacobson radical of the ring R. Then R is adjoint regularif and only if R/J is adjoint regular, idempotents of R/J can be lift, and eJe = 0 for anyidempotent e∈R.Theorem 3.4.2 Let R = S +T, where S,T are both adjoint regular subrings of R. If anyelement of ST is quasi-regular in R, then R is adjoint regular if and only if eJ(S)f = eJ(T)f = 0for any idempotents e,f∈S∪T, where J(S),J(T) are the Jacobson radicals of S,T, respectively.Corollary 3.4.2 Let R be a sum of a radical subring K with an adjoint regular subringS. If any element of KS is quasi-regular in R, then R is adjoint regular if and only if eKe = 0for any idempotent e∈S.