Ordered left PP monoids

2014 ◽  
Vol 64 (6) ◽  
Author(s):  
Xiaoping Shi
Keyword(s):  
Left Ideal ◽  
Ordered Monoid ◽  
Green Relations

AbstractAn ordered monoid S in which every principal left ideal, regarded as an S-poset, is projective is called an ordered left PP monoid, for short, an ordered lpp monoid. In this paper, we introduce a new kind of ordered relations instead of Green relations adopted by J. B. Fountain, ordered lpp monoids are described by these ordered relations. Some similar results of c-rpp semigroups are duduced and proved, in particular, Fountain’s results about C-lpp monoids are generalized to ordered monoids.

scholarly journals ON ANNIHILATOR IDEALS OF SKEW MONOID RINGS

2009 ◽  
Vol 52 (1) ◽  
pp. 161-168 ◽  
Author(s):  
LIU ZHONGKUI ◽  
YANG XIAOYAN
Keyword(s):  
Left Ideal ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Monoid Ring ◽  
Ordered Monoid ◽  
Monoid Homomorphism ◽  
Necessary And Sufficient ◽  
Left Annihilator

AbstractA ring R is called a left APP-ring if the left annihilator lR(Ra) is pure as a left ideal of R for every a ∈ R; R is called (left principally) quasi-Baer if the left annihilator of every (principal) left ideal of R is generated by an idempotent. Let R be a ring and M an ordered monoid. Assume that there is a monoid homomorphism φ: M ⟶ Aut(R). We give a necessary and sufficient condition for the skew monoid ring (induced by φ) to be left APP (left principally quasi-Baer, quasi-Baer, respectively).

scholarly journals Noetherian properties in composite generalized power series rings

2020 ◽  
Vol 18 (1) ◽  
pp. 1540-1551
Author(s):  
Jung Wook Lim ◽  
Dong Yeol Oh
Keyword(s):  
Necessary Conditions ◽  
Sufficient Conditions ◽  
Commutative Rings ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Generalized Power Series ◽  
Ordered Monoid ◽  
Power Series Rings ◽  
Necessary And Sufficient ◽  
Equivalent Conditions

Abstract Let ({\mathrm{\Gamma}},\le ) be a strictly ordered monoid, and let {{\mathrm{\Gamma}}}^{\ast }\left={\mathrm{\Gamma}}\backslash \{0\} . Let D\subseteq E be an extension of commutative rings with identity, and let I be a nonzero proper ideal of D. Set \begin{array}{l}D+[\kern-2pt[ {E}^{{{\mathrm{\Gamma}}}^{\ast },\le }]\kern-2pt] := \left\{f\in [\kern-2pt[ {E}^{{\mathrm{\Gamma}},\le }]\kern-2pt] \hspace{0.15em}|\hspace{0.2em}f(0)\in D\right\}\hspace{.5em}\text{and}\\ \hspace{0.2em}D+[\kern-2pt[ {I}^{{\Gamma }^{\ast },\le }]\kern-2pt] := \left\{f\in [\kern-2pt[ {D}^{{\mathrm{\Gamma}},\le }]\kern-2pt] \hspace{0.15em}|\hspace{0.2em}f(\alpha )\in I,\hspace{.5em}\text{for}\hspace{.25em}\text{all}\hspace{.5em}\alpha \in {{\mathrm{\Gamma}}}^{\ast }\right\}.\end{array} In this paper, we give necessary conditions for the rings D+[\kern-2pt[ {E}^{{{\mathrm{\Gamma}}}^{\ast },\le }]\kern-2pt] to be Noetherian when ({\mathrm{\Gamma}},\le ) is positively ordered, and sufficient conditions for the rings D+[\kern-2pt[ {E}^{{{\mathrm{\Gamma}}}^{\ast },\le }]\kern-2pt] to be Noetherian when ({\mathrm{\Gamma}},\le ) is positively totally ordered. Moreover, we give a necessary and sufficient condition for the ring D+[\kern-2pt[ {I}^{{\Gamma }^{\ast },\le }]\kern-2pt] to be Noetherian when ({\mathrm{\Gamma}},\le ) is positively totally ordered. As corollaries, we give equivalent conditions for the rings D+({X}_{1},\ldots ,{X}_{n})E{[}{X}_{1},\ldots ,{X}_{n}] and D+({X}_{1},\ldots ,{X}_{n})I{[}{X}_{1},\ldots ,{X}_{n}] to be Noetherian.

scholarly journals Duality and the existence of weakly completely continuous elements in aB*-algebra

1972 ◽  
Vol 13 (1) ◽  
pp. 56-60 ◽  
Author(s):  
B. J. Tomiuk
Keyword(s):  
Hilbert Space ◽  
Left Ideal ◽  
Linear Operators ◽  
Completely Continuous ◽  
Identity Modulo

Ogasawara and Yoshinaga [9] have shown that aB*-algebra is weakly completely continuous (w.c.c.) if and only if it is*-isomorphic to theB*(∞)-sum of algebrasLC(HX), where eachLC(HX)is the algebra of all compact linear operators on the Hilbert spaceHx.As Kaplansky [5] has shown that aB*-algebra isB*-isomorphic to theB*(∞)-sum of algebrasLC(HX)if and only if it is dual, it follows that a5*-algebraAis w.c.c. if and only if it is dual. We have observed that, if only certain key elements of aB*-algebraAare w.c.c, thenAis already dual. This observation constitutes our main theorem which goes as follows.A B*-algebraAis dual if and only if for every maximal modular left idealMthere exists aright identity modulo M that isw.c.c.

scholarly journals Simplex algebras and their representation

1973 ◽  
Vol 14 (2) ◽  
pp. 136-144
Author(s):  
M. S. Vijayakumar
Keyword(s):  
Left Ideal ◽  
Maximal Ideal ◽  
Banach Algebras ◽  
Principal Component ◽  
Nonempty Interior ◽  
Regular Elements ◽  
Maximal Left Ideal ◽  
The Relationship ◽  
Order Identity ◽  
Real Matrices

This paper establishes a relationship (Theorem 4.1) between the approaches of A. C. Thompson [8, 9] and E. G. Effros [2] to the representation of simplex algebras, that is, real unital Banach algebras that are simplex spaces with the unit for order identity. It proves that the (nonempty) interior of the associated cone is contained in the principal component of the set of all regular elements of the algebra. It also conjectures that each maximal ideal (in the order sense—see below) of a simplex algebra contains a maximal left ideal of the algebra. This conjecture and other aspects of the relationship are illustrated by considering algebras of n × n real matrices.

scholarly journals A note on extensions of Baer and P. P. -rings

1974 ◽  
Vol 18 (4) ◽  
pp. 470-473 ◽  
Author(s):  
Efraim P. Armendariz
Keyword(s):  
Left Ideal ◽  
Polynomial Identity ◽  
Polynomial Rings ◽  
Nilpotent Elements ◽  
Baer Ring ◽  
Baer Rings ◽  
Left Annihilator

Baer rings are rings in which the left (right) annihilator of each subset is generated by an idempotent [6]. Closely related to Baer rings are left P.P.-rings; these are rings in which each principal left ideal is projective, or equivalently, rings in which the left annihilator of each element is generated by an idempotent. Both Baer and P.P.-rings have been extensively studied (e.g. [2], [1], [3], [7]) and it is known that both of these properties are not stable relative to the formation of polynomial rings [5]. However we will show that if a ring R has no nonzero nilpotent elements then R[X] is a Baer or P.P.-ring if and only if R is a Baer or P.P.-ring. This generalizes a result of S. Jøndrup [5] who proved stability for commutative P.P.-rings via localizations – a technique which is, of course, not available to us. We also consider the converse to the well-known result that the center of a Baer ring is a Baer ring [6] and show that if R has no nonzero nilpotent elements, satisfies a polynomial identity and has a Baer ring as center, then R must be a Baer ring. We include examples to illustrate that all the hypotheses are needed.

Structure Theory of Complemented Banach Algebras

1962 ◽  
Vol 14 ◽  
pp. 651-659 ◽  
Author(s):  
Bohdan J. Tomiuk
Keyword(s):  
Hilbert Space ◽  
Left Ideal ◽  
Hilbert Spaces ◽  
Banach Algebras ◽  
Inner Product ◽  
Structure Theory ◽  
Orthogonal Complement

If A is an H*-algebra, then the orthogonal complement of a closed right (left) ideal I is a closed right (left) ideal P. Saworotnow (7) considered Banach algebras which are Hilbert spaces and in which the closed right ideals satisfy the complementation property of an H*-algebra. In our right complemented Banach algebras we drop the requirement of the existence of an inner product and only assume that for every closed right ideal I there is a closed right ideal IP which behaves like the orthogonal complement in a Hilbert space (Definition 1). Thus our algebras may be considered as a generalization of Saworotnow's right complemented algebras.

Stable Rings

1980 ◽  
Vol 23 (2) ◽  
pp. 173-178 ◽  
Author(s):  
S. S. Page
Keyword(s):  
Left Ideal ◽  
Associative Ring ◽  
Commutative Rings ◽  
Von Neumann ◽  
Goldie Dimension ◽  
Von Neumann Regular

Let R be an associative ring with identity. If R is von- Neumann regular of a left v-ring, then for each left ideal, I, we have I2 = I. In this note we study rings such that for each left ideal L there exists an integer n = n(L)>0 such that Ln = Ln+1. We call such rings stable rings. We completely describe the stable commutative rings. These descriptions give rise to concepts related to, but more general than, finite Goldie dimension and T-nilpotence, and a notion of power pure.

On Very Large One Sided Ideals of a Ring

1966 ◽  
Vol 9 (2) ◽  
pp. 191-196 ◽  
Author(s):  
Kwangil Koh
Keyword(s):  
Left Ideal ◽  
Prime Ring ◽  
Cardinal Number ◽  
Singular Element

If R is a ring, a right (left) ideal of R is said to be large if it has non-zero intersection with each non-zero right (left) ideal of R [8]. If S is a set, let |S| be the cardinal number of S. We say a right (left) ideal I of a ring R is very large if |R/I| < < No. If a is an element of a ring R such that (a)r = {r ∊ R|ar = 0} is very large then we say a is very singular. The set of all very singular elements of a ring R is a two sided ideal of R. If R is a prime ring, then 0 is the only very singular element of R and a very large right (left) ideal of R is indeed large provided that R is not finite.

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