A. Anjaneyulu

Bio: A. Anjaneyulu is an academic researcher from Acharya Nagarjuna University. The author has contributed to research in topics: Semiprime ring & Prime ideal. The author has an hindex of 3, co-authored 3 publications receiving 45 citations.

Semigroups in which prime ideals are maximal

Abstract: In this paper, the terms, Maximal Γ-ideal, primary Γ-semigroup and Prime Γ-ideal are introduced. It is proved that if S is a Γsemigroup with identity and if ( non zero, assume this if S has zero) proper prime Γ-ideals in S are maximal then S is primary Γ-semigroup. Also it is proved that if S is a right cancellative quasi commutative Γ-emigroup and if S is a primary Γsemigroup or a Γsemigroup in which semiprimary Γideals are primary, then for any primary Γ-ideal Q, √ Q is non-maximal implies Q = √ Q is prime. It is proved that if S is a right cancellative quasi commutative Γ-semigroup with identity, then 1) Proper prime Γ-ideals in S are maximal. 2) S is a primary Γ-semigroup. 3) Semiprimary Γ-ideals in S are primary, 4) If x and y are not units in S, then there exists natural numbers n and m such that (x Γ) n-1 x = yΓs and (yΓ) m-1 y = xΓr. For some s, r ∈ S are equivalent. Also it is proved that if S is a duo Γ–semigroup with identity, then 1) Proper prime Γ– ideals in S are maximal. 2) S is either a Γ– group and so Archimedian or S has a unique prime Γ–ideal P such that S = G∪ P, where G is the Γ–group of units in S and P is an Archimedian sub Γ–semi group of S are equivalent. In either case S is a primary Γ–semigroup and S has atmost one idempotent different from identity. It is proved that if S is a duo Γ-semigroup without identity, then S is a primary Γ-semigroup in which proper prime Γ-ideals are maximal if and only if S is an Archimedian Γ-semigroup. It is also proved that if S is a quasi commutative Γ-semigroup containing cancellable elements, then 1) The proper prime Γ-ideals in S are maximal. 2) S is a Γ-group or S is a cancellative Archimedian Γ-semigroup not containing identity or S is an extension of an Archimedian Γ-semigroup by a Γ-group S containing an identity are equivalent. Mathematical subject classification (2010): 20M07; 20M11; 20M12.

Semigroups in which prime ideals are maximal

Abstract: In this paper, the terms, Maximal Γ-ideal, primary Γ-semigroup and Prime Γ-ideal are introduced. It is proved that if S is a Γsemigroup with identity and if ( non zero, assume this if S has zero) proper prime Γ-ideals in S are maximal then S is primary Γ-semigroup. Also it is proved that if S is a right cancellative quasi commutative Γ-emigroup and if S is a primary Γsemigroup or a Γsemigroup in which semiprimary Γideals are primary, then for any primary Γ-ideal Q, √ Q is non-maximal implies Q = √ Q is prime. It is proved that if S is a right cancellative quasi commutative Γ-semigroup with identity, then 1) Proper prime Γ-ideals in S are maximal. 2) S is a primary Γ-semigroup. 3) Semiprimary Γ-ideals in S are primary, 4) If x and y are not units in S, then there exists natural numbers n and m such that (x Γ) n-1 x = yΓs and (yΓ) m-1 y = xΓr. For some s, r ∈ S are equivalent. Also it is proved that if S is a duo Γ–semigroup with identity, then 1) Proper prime Γ– ideals in S are maximal. 2) S is either a Γ– group and so Archimedian or S has a unique prime Γ–ideal P such that S = G∪ P, where G is the Γ–group of units in S and P is an Archimedian sub Γ–semi group of S are equivalent. In either case S is a primary Γ–semigroup and S has atmost one idempotent different from identity. It is proved that if S is a duo Γ-semigroup without identity, then S is a primary Γ-semigroup in which proper prime Γ-ideals are maximal if and only if S is an Archimedian Γ-semigroup. It is also proved that if S is a quasi commutative Γ-semigroup containing cancellable elements, then 1) The proper prime Γ-ideals in S are maximal. 2) S is a Γ-group or S is a cancellative Archimedian Γ-semigroup not containing identity or S is an extension of an Archimedian Γ-semigroup by a Γ-group S containing an identity are equivalent. Mathematical subject classification (2010): 20M07; 20M11; 20M12.

Partially Ordered Γ- Semigroups

Abstract: In this paper, the notion of an ordered Γ-semigroup is introduced and some examples are given. Further the terms commutative ordered Γ-semigroup, quasi commutative ordered Γ-semigroup, normal ordered Γsemigroup, left pseudo commutative ordered Γsemigroup, right pseudo commutative ordered Γsemigroup are introduced. It is proved that (1) if S is a commutative ordered Γ-semigroup then S is a quasi commutative ordered Γ-semigroup, (2) if S is a quasi commutative ordered Γ-semigroup then S is a normal ordered Γ-semigroup, (3) if S is a commutative ordered Γ-semigroup, then S is both a left pseudo commutative and a right pseudo commutative ordered Γ-semigroup. Further the terms; left identity, right identity, identity, left zero, right zero, zero of an ordered Γ-semigroup are introduced. It is proved that if a is a left identity and b is a right identity of an ordered Γ-semigroup S, then a = b. It is also proved that any ordered Γsemigroup S has at most one identity. It is proved that if a is a left zero and b is a right zero of an ordered Γsemigroup S, then a = b and it is also proved that any ordered Γ-semigroup S has at most one zero element. The terms; ordered Γ-subsemigroup, ordered Γsubsemigroup generated by a subset, α-idempotent, Γ-idempotent, strongly idempotent, midunit, r-element, regular element, left regular element, right regular element, completely regular element, (α, β)-inverse of an element, semisimple element and intra regular element in an ordered Γ-semigroup are introduced. Further the terms idempotent ordered Γ-semigroup and generalized commutative ordered Γ-semigroup are introduced. It is proved that every α-idempotent element of an ordered Γ-semigroup is regular. It is also proved that, in an ordered Γ-semigroup, a is a regular element if and only if a has an ( , )-inverse. It is proved that, (1) if a is a completely regular element of an ordered Γ-semigroup S, then a is both left regular and right regular, (2) if „a‟ is a completely regular element of an ordered Γ-semigroup S, then a is regular and semisimple, (3) if „a‟ is a left regular element of an ordered Γ-semigroup S, then a is semisimple, (4) if „a‟ is a right regular element of an ordered Γ-semigroup S, then a is semisimple, (5) if „a‟ is a regular element of an ordered Γsemigroup S, then a is semisimple and (6) if „a‟ is a intra regular element of an ordered Γsemigroup S, then a is semisimple. The term separative ordered Γ-semigroup is introduced and it is proved that, in a separative ordered Γ-semigroup S, for any x, y, a, b ∈ S, the statements (i) xΓa ≤ xΓb if and only if aΓx ≤ bΓx, (ii) xΓ xΓa ≤ xΓxΓb implies xΓa ≤ xΓb,(iii) xΓyΓa ≤ xΓyΓb implies yΓxΓa ≤ yΓxΓb hold.

Left (Right) Regular and Transposition Regular Semigroups and Their Structures

Abstract: Regular semigroups and their structures are the most wonderful part of semigroup theory, and the contents are very rich. In order to explore more regular semigroups, this paper extends the relevant classical conclusions from a new perspective: by transforming the positions of the elements in the regularity conditions, some new regularity conditions (collectively referred to as transposition regularity) are obtained, and the concepts of various transposition regular semigroups are introduced (L1/L2/L3, R1/R2/R3-transposition regular semigroups, etc.). Their relations with completely regular semigroups and left (right) regular semigroups, proposed by Clifford and Preston, are analyzed. Their properties and structures are studied from the aspects of idempotents, local identity elements, local inverse elements, subsemigroups and so on. Their decomposition theorems are proved respectively, and some new necessary and sufficient conditions for semigroups to become completely regular semigroups are obtained.

Prime Radicals in Ternary Semigroups

Abstract: In this paper the terms completely prime ideal, prime ideal, m-system. globally idempotent , semi simple elements of a ternary semigroup are Introduced. It is proved that an ideal A of a ternary semigroup T is completely prime if and only if T\A is either sub semigroup of T or empty. It is proved that if T is a globally idempotent ternary semigroup then every maximal ideal of T is a prime ideal of T. In this paper the terms completely semiprime ideal, semiprime ideal, n-system, d-system and i-system are introduced. It is proved that the non-empty intersection of any family of a completely prime ideal and prime ideal of a ternary semigroup T is a completely semiprime ideal of T. It is also proved that an ideal A of a ternary semigroup T is completely semiprime if and only if T\A is a d-system of T or empty. It is proved that if N is an n-system in a ternary semigroup T and a