| There is an interest in the relationship among a ring and its adjoint groupand semigroup, which has been described by many authors. For example, it iswell-known that a ring R is a Jacobson radical if and only if its adjoint semigroup(R,?) is a group.In 1942, under suggestion of M. Hall, Perlis introduced the notion of quasiregu-lar elements to describe the radical of an algebra. For an element x in an algebraA, if there exists y∈A such that x + xy + y = 0, then x is called quasiregu-lar. In 1945, Jacobson extended the idea to general rings and introduced circlemultiplication defined as follows.Jacobson observed that (R,?) is a monoid with identity 0 (the additive zero).(R,(?)), abbreviated, R(?), is called the circle semigroup or adjoint semigroup of R.The group of units in R(?) is called the circle group or adjoint group of R.For a ring with 1, its adjoint semigroup and multiplicative semigroup are iso-morphic, and so they are the same in essential. However, there is a big di?erencebetween them for a ring without 1. For example, if R is a zero ring (i.e., R2 = 0),R(?) is a group, but the multiplication semigroup R is zero semigroup. Thoughany ring R can be embedded in a unitary ring R1 as an ideal and in this case R(?)is an subsemigroup of the multiplicative semigroup of R1, R1 and its multiplica-tive semigroup are usually too complex to study. Hence the investigation on the adjoint semigroup has its independent interest. The goals are to characterize theadjoint group and semigroup of a given ring, and conversely, to characterize therings with special adjoint groups and semigroups.Many papers are devoted to describe the relationship among a ring and itsadjoint group and semigroup and to study generalizations of the circle multipli-cations. Several research works have been applied to modular group algebras, forexample.In this dissertation we are concerned with a generalization of circle multipli-cation, called generalized adjoint multiplication, abbreviated, GA-multiplication,which is a binary operation on R satisfying the following conditions.(1) associative law(2) generalized distributive laws(3) compatibilityThe semigroup (R, ), abbreviated R , is called GA-semigroup of R. The maingoal of this dissertation is to describe the GA-semigroups of a given ring andconversely to describe the rings with a given GA-semigroup.We first characterize GA-multiplications in terms of bitranslations and char-acterize the GA-semigroup with zero (identity) as the multiplicative (adjoint)semigroup. It is proved that GA-semigroups of aπn-regular ring areπn-regular.We relate existence of GA-semigroups with idempotents to lifting idempotents,and construct GA-semigroups with idempotents by means of Morita context. Weconclude that if the multiplication semigroup is a P semigroup, then so is ev-ery GA-semigroup, and if a GA- semigroup is a P semigroup, then so is R?,where P stands for orthodox, right inverse, inverse, pseudo-inverse, E-unitaryand completely simple.The main results of this dissertation are as follows. Theorem 2.1.1 Let (θ,(?)) be an associated pair of a ring R and definefor any x,y∈R. Then is a generalized adjoint multiplication on R induced by(θ,(?)). Conversely, every generalized adjoint multiplication on R can be obtainedin this fashion by setting (?) = 0 0,θx = 0 x (?) 0 0 and xθ= x 0 (?) 0 0.Moreover, the correspondence (θ,(?))→is a 1 (?) 1 correspondence between theassociated pairs of R and generalized adjoint multiplications on R.Theorem 2.1.3 LetR be a GA-semigroup of a ring R. Then(i) R has identity if and only if R R(?);(ii) R has zero if and only if R R ;(iii) if R has identity,then R R R(?).Theorem 2.2.1 For a non-negative integer n,if a ring R is(left,right,completely)πn-regular, then so is its any GA-semigroup.Theorem 2.3.2 Consider the following conditions:(i) every GA-semigroup of R contains(central)idempotents;(ii) in any ideal extension R(?) of R, idempotents of R(?)/R can be lifted to idem-potents of R(?) (contained in the centralizer of R in R(?));(iii) idempotents of (?)(R)/π(R) can be lifted to idempotents of (?)(R) (containedinΓ(R)). Then (iii) (?) (i) (?) (ii). Moreover, if Ann(R) = 0, then (i), (ii) and(iii) are equivalent.Theorem 2.3.5 Let R be a ring with descending chain condition for princi-pal right ideals. Then any GA-semigroup of R is completelyπ-regular. Particu-larly,any GA-semigroup of a right Artinian ring is completelyπ-regular.Theorem 2.4.1 R has a central idempotent if and only if R R0(?)×R1(?) forsome ideals R0 and R1 of R such that R = R0⊕R1.Theorem 2.4.2 Any GA-semigroup R of a strongly regular ring containscentral idempotents, and so R R0×R1 for some ideals R0 and R1 of R such that R = R0⊕R1.Theorem 3.1.1 Let R = M(S,T,U,V ) with V U = 0. Then the E11-GA-semigroup R is regular if and only if S is an adjoint regular ring, T is a regularring and E(S)U = V E(S) = 0, and if so, then(i) R is an adjoint regular ring;Theorem 3.2.1 The following statements are equivalent for a GA-semigroupR of R.(i) R is orthodox;(ii) R is a union of groups and E(R ) is a regular band;(iii) R M11(S,T,U,V ), where S? is an orthodox semigroup, T is a stronglyregular ring, and E(S)U = V E(S) = UV = V U = 0.Theorem 3.2.2 The following statements are equivalent for a ring R.(i) R has an orthodox GA-semigroup;(ii) R? is an orthodox semigroup;(iii) R? is a union of groups and E(R?) is a regular band.Theorem 3.3.1 A ring R has a right inverse GA-semigroup if and only if R?is right inverse. Moreover, a GA-semigroup R of R is right inverse if and only ifR M11(S,T,U,0), where S? is right inverse, T is a strongly regular ring, andE(S)U = 0.Theorem 3.3.2 The following statements are equivalent for a GA-semigroupR of R.(i) R is inverse;(ii) R is a regular semigroup in which idempotents are all central;(ii) R R0?×R1?, where R0 and R1 are ideals of R such that R = R0⊕R1, R0 is a strongly regular ring and R1? is inverse.Theorem 3.3.4 A GA-semigroup R of a ring is a pseudoinverse semigroupif and only if R M11(S,T,U,V ), where S? is inverse, T is a strongly regularring, and E(S)U = V E(S) = UT = TV = V U = 0.Theorem 3.3.5 The following statements for a ring R are equivalent.(i) R has a pseudoinverse GA-semigroup;(ii) R has an inverse GA-semigroup;(iii) R? is inverse.Theorem 3.3.6 Every GA-semigroup of R is inverse (orthodox, pseudoin-verse)if and only if R is a strongly regular ring.Theorem 3.4.1 A GA-semigroup R of a ring R is E-unitary if and only ifR M11(S,T,U,V ), where S is a direct sum of a Boolean ring with a radicalring, T is a Boolean ring, and E(S)U = V E(S) = UT = TV = UV = V U = 0.Theorem 3.4.2 The following conditions are equivalent for a ring R.(i) R has an E-unitary GA-semigroup;(ii) R is a direct sum of a Boolean ring with a radical ring;(iii) R? is E-unitary.Theorem 3.5.1 A GA-semigroup R of R is a completely simple semigroupif and only if R M11(S,0,U,V ), where S is a radical ring.We conclude this dissertation with the following remarkR has the propertyP ? R has the propertyP;? R?has the propertyP,where P stands for orthodox, right inverse, inverse, pseudoinverse, E-unitary,and completely simple, respectively. |
