Minimal Ideal Research Articles

Ascending chain conditions on right ideals of semigroups

ABSTRACT We call a semigroup S right noetherian if it satisfies the ascending chain condition on right ideals, and we say that S satisfies ACCPR if it satisfies the ascending chain condition on principal right ideals. We investigate the behavior of these two conditions with respect to ideals and ideal extensions, with a particular focus on minimal and 0-minimal one-sided ideals. In particular, we show that the property of satisfying ACCPR is inherited by right and left ideals. On the other hand, we exhibit an example of a right noetherian semigroup with a minimal ideal that is not right noetherian.

The Characteristic of the Minimal Ideals and the Minimal Generalized Ideals in Rings

In this paper, we prove that the characteristic of a minimal ideal and a minimal generalized ideal, which is meant to be one of minimal left ideal, minimal right ideal, bi‐ideal, quasi‐ideal, and (m, n)‐ideal in a ring, is either zero or a prime number p. When the characteristic is zero, then the minimal ideal (minimal generalized ideal) as additive group is torsion‐free, and when the characteristic is p, then every element of its additive group has order p. Furthermore, we give some properties for minimal ideals and for generalized ideals which depend on their characteristics.

On minimal ideals in pseudo-finite semigroups

AbstractA semigroupSis said to be right pseudo-finite if the universal right congruence can be generated by a finite set$U\subseteq S\times S$, and there is a bound on the length of derivations for an arbitrary pair$(s,t)\in S\times S$as a consequence of those inU. This article explores the existence and nature of a minimal ideal in a right pseudo-finite semigroup. Continuing the theme started in an earlier work by Dandan et al., we show that in several natural classes of monoids, right pseudo-finiteness implies the existence of a completely simple minimal ideal. This is the case for orthodox monoids, completely regular monoids, and right reversible monoids, which include all commutative monoids. We also show that certain other conditions imply the existence of a minimal ideal, which need not be completely simple; notably, this is the case for semigroups in which one of the Green’s preorders${\leq _{\mathcal {L}}}$or${\leq _{\mathcal {J}}}$is left compatible with multiplication. Finally, we establish a number of examples of pseudo-finite monoids without a minimal ideal. We develop an explicit construction that yields such examples with additional desired properties, for instance, regularity or${\mathcal {J}}$-triviality.

The socle of Leavitt path algebras over a semiprime ring

The Reduction Theorem in Leavitt path algebra over a commutative unital ring is very important to prove that the Leavitt path algebra is semiprime if and only if the ring is also semiprime. Any minimal ideal in the semiprime ring and line point will construct a left minimal ideal in the Leavitt path algebra. Vice versa, any left minimal ideal in the semiprime Leavitt path algebra can be found both minimal ideal in the semiprime ring and line point that generate it. The socle of semiprime Leavitt path algebra is constructed by minimal ideals of the semiprime ring and the set of all line points.

Minimal Ideals and Primitivity in Near-rings

We address and answer the question when a minimal ideal of a zero symmetric near-ring is a primitive near-ring. This implies that a minimal ideal of a zero symmetric near-ring is a simple near-ring in many natural situations.

MINIMAL IDEALS OF ABEL-GRASSMANN GROUPOIDS

The properties of Abel-Grassmann groupoids have been attracted the attention of many authors. The aim of this paper is to study the properties of the minimal left ideals of an Abel-Grassmann groupoid ( in brevity, an AG- groupoid ) with left identity. It is proved that if L is a minimal left ideal of an AG-groupoid S with left identity then Lc is a minimal left ideal of S for all c ∈ S. We also show that the kernel K of an AG-groupoid S ( the intersection of all two sided ideals of S if exists)is simple and the class sumof all minimal left ideals of S containing at least one minimal left ideal of S is precisly the kernel K of S. Finally, we show that if S is an AG-groupoid with left identity then Sa 2 S = Sa 2 for all a ∈ S. Finally, if S is an AG-groupoid with left identity and does not contain any non-trivial nilpotent ideals, then every minimal ideal of S is simple.A number of classical results of L. M. Gluskin and O. Steinfeld given in 1978 (3) concerning the minimal one sided ideals of semigroups and rings are consequently extended to and strengthened in AG-groupoids.

Trivial minimal ideals of Jordan systems

We show that in Jordan systems (algebras, triple systems, and pairs) monomials containing two elements of a trivial minimal ideal vanish, so improving the answer given by Anquela and Cortés in [J.A. Anquela, T. Cortés, Minimal ideals of Jordan systems, Invent. Math. 168 (2007) 83–90] to the problem of Nam and McCrimmon in [N.S. Nam, K. McCrimmon, Minimal ideals in quadratic Jordan algebras, Proc. Amer. Math. Soc. 88 (4) (1983) 579–583], inspired by the problem posed in 1971 by Zhevlakov in the Dniester Notebook: Unsolved Problems in the Theory of Rings and Modules (second ed., 1976) (see Filippov et al. (1993) [5]).

Extending Polynomials in Maximal and Minimal Ideals

Given a homogeneous polynomial on a Banach space E belonging to some maximal or minimal polynomial ideal, we consider its iterated extension to an ultrapower of E and prove that this extension remains in the ideal and has the same ideal norm. As a consequence, we show that the Aron–Berner extension is a well defined isometry for any maximal or minimal ideal of homogeneous polynomials. This allows us to obtain symmetric versions of some basic results of the metric theory of tensor products.

Flat orbits, minimal ideals and spectral synthesis

Let G = exp \({\mathfrak{g}}\) be a connected, simply connected, nilpotent Lie group and let ω be a continuous symmetric weight on G with polynomial growth. In the weighted group algebra \({L^{1}_{\omega}(G)}\) we determine the minimal ideal of given hull \({\{\pi_{l'} \in \hat{G} | l' \in l + \mathfrak{n}^{\perp}\}}\), where \({\mathfrak{n}}\) is an ideal contained in \({\mathfrak{g}(l)}\), and we characterize all the L∞(G/N)-invariant ideals (where \({N = {\rm exp}\, \mathfrak{n}}\)) of the same hull. They are parameterized by a set of G-invariant, translation invariant spaces of complex polynomials on N dominated by ω and are realized as kernels of specially built induced representations. The result is particularly simple if the co-adjoint orbit of l is flat.

ON SUPERNILPOTENT NONSPECIAL RADICALS

AbstractLet ρ be a supernilpotent radical. Let ρ* be the class of all rings A such that either A is a simple ring in ρ or the factor ring A/I is in ρ for every nonzero ideal I of A and every minimal ideal M of A is in ρ. Let $\mathcal {L}\left ( \rho ^{\ast }\right ) $ be the lower radical determined by ρ* and let ρφ denote the upper radical determined by the class of all subdirectly irreducible rings with ρ-semisimple hearts. Le Roux and Heyman proved that $\mathcal {L}\left ( \rho ^{\ast }\right ) $ is a supernilpotent radical with $\rho \subseteq \mathcal {L}\left ( \rho ^{\ast }\right ) \subseteq \rho _{\varphi }$ and they asked whether $\mathcal {L} \left ( \rho ^{\ast }\right ) =\rho _{\varphi }$ if ρ is replaced by β, ℒ , 𝒩 or 𝒥 , where β, ℒ , 𝒩 and 𝒥 denote the Baer, the Levitzki, the Koethe and the Jacobson radical, respectively. In the present paper we will give a negative answer to this question by showing that if ρ is a supernilpotent radical whose semisimple class contains a nonzero nonsimple * -ring without minimal ideals, then $\mathcal {L}\left ( \rho ^{\ast }\right ) $ is a nonspecial radical and consequently $\mathcal {L}\left ( \rho ^{\ast }\right ) \neq \rho _{\varphi }$. We recall that a prime ring A is a * -ring if A/I is in β for every $0\neq I\vartriangleleft A$.

Representations of finitary Lie algebras

Let F be an algebraically closed field of characteristic ≠2,3, W a F -vector space and L⊂ gl (W) with nil W L =(0), dim F L =∞. Suppose L⊂ fgl (W) is a finitary subalgebra. The faithful irreducible L -modules are determined. It is shown that L has minimal ideals. If a minimal ideal S is infinite-dimensional then SW is a completely reducible L -module. Suppose L∩ fgl (W)≠(0) , W is L -irreducible and char( F )>3. Then L is classified in terms of L∩ fgl (W) .

A construction of a compact right topological semigroup

It is well-known that the structure of βℕ, the Stone—Čech compactification of the discrete semigroup (ℕ, +), is very complex. For example, it has 2c minimal left ideals and 2c minimal right ideals, its minimal ideal contains 2c copies of the free group on 2c generators, see [5], and Lisan[8] proved the existence of 2c copies of the same group outside the closure of the minimal ideal.

Minimal ideals in near-rings and localized distributivity conditions

AbstractLetB, S, andTbe subsets of a (left) near-ringRwithBandTnonempty. We sayBis (S, T)-distributiveifs(b1+b2)t=sb1t+sb2t, for eachs∈S,b1,2∈B,t∈T. Basic properties for this type of ‘localized distributivity’ condition are developed, examples are given, and applications are made in determining the structure of minimal ideals.Theorem. IfIis a minimal ideal ofRandIkis (Im,In)-distributive for somek,n≧ 1,m≧ 0, then eitherI2= 0 orIis a simple, nonnilpotent ring with every element ofIdistributive inR. Theorem. LetRkbe (Rm,Rn)-distributive, for somek,n≧ 1, m ≧ 0; ifRis semiprime or is a subdirect product of simple near-rings, thenRis a ring. Connections are established with near-rings which satisfy a permutation identity and with weakly distributive near-rings. IfR→A→ 0 is an exact sequence of near-rings, then conditions onAare given which will impose conditions on the minimal ideals ofR.

Minimal ideals in quadratic Jordan algebras

In associative and alternative algebras a minimal ideal is either trivial or simple. This is not known for quadratic Jordan algebras. In the present note we show that a minimal ideal is either trivial or D \mathcal {D} -simple (possesses no proper ideals invariant under all inner derivations induced from the ambient algebra). In particular, the heart of any quadratic Jordan algebra is either trivial or D \mathcal {D} -simple. Hearts have recently played an important role in Zelmanov’s theory of prime Jordan algebras.

On the design of mean-square error channel coding systems using cyclic codes

Channel coding systems that employ linear block codes can be designed according to a mean-square error criterion. The optimum encoder and decoder pair is determined by selecting special elements in a generalized frequency domain based upon the values of key parameters called ratio weights. Systems may be designed that employ either hard or soft decision variables in the decoder. When cyclic codes are used, a minimal ideal decomposition of the code space shows that the ratio weights possess constant values on certain subsets in the frequency domain. This permits a drastic reduction in the number of ratio weights that need to be computed. For a cyclic code with <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">t</tex> minimal ideals, this design procedure narrows the search for constancy subsets of the ratio weights to <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(2^{t}-1)</tex> subsets of the general space. Only one ratio weight needs to be computed for each constancy subset.

Complements of Minimal Ideals in Solvable Lie Rings

AbstractConditions for the existence and conjugacy of complements of certain minimal ideals of solvable Lie algebras over a Noetherian ring R are considered. Let L be a solvable Lie algebra and A be a minimal ideal of L. If L/A is nilpotent and L is not nilpotent then A has a complement in L, all such complements are conjugate and self-normalizing and if C is a complement then there exists an x∈L such that C = {y∈L; yadnx = 0 for some n = 1, 2,…}. A similar result holds if A is self-centralizing and a finitely generated R-module.

The homotopic type of certain compact semigroups

The relation between homotopic types of compact linearly connected semigroups and their minimal ideals is investigated. In Theorem 1 we prove that, if a semigroup has a homotopic unit, even if it is only one-sided, then the minimal ideal is a deformation retract of the semigroup and so is homotopically equivalent to the semigroup. In Theorem 2 it is proved that a Q-semigroup is weakly homotopically equivalent to its minimal ideal.

The socle of an alternative ring

The main facts about the minimal ideals and minimal right ideals of an associative ring are well known. In this paper we prove corresponding results for an alternative ring R. We make no restriction on the characteristic of R, but will often impose restrictions of semiprimeness type. (R is semiprime provided it has no ideal T such that T # (0) = P). Throughout this paper “ring” will mean “alternative ring”, and R will be a ring. We write A < R (A <, R; A <, R) to denote that A is an ideal (right ideal, left ideal) of R. If A is a minimal element of the set {M : (0) # M < R} (respectively, {M : (0) # M <,. R}), partially ordered by inclusion, we say that A is a minimal ideal (respectively, minimal fight ideal) of R, and write A <, R (respectively, A <,, R). Similarly for A <,, R. In Section 2, we show that if A <, R then either A2 = (0) or A is simple; this result is due in part to Zhevlakov. In Section 3 we similarly characterize A if A &. R. Roughly, if A2 # (0), then A is a minimal right ideal of the ideal C it generates in R, and, in general, C is simple. We also show that A is of the form eR, where e is a nuclear idempotent. In Section 4 we consider the right socle of R : S,(R) = Z(M : M <,, R). Under a suitable weak condition S,(R) is a two-sided ideal, and under a slightly stronger condition (weaker than semiprimeness of R) it coincides with the analogously defined left socle S,(R). In this situation we define the socle of R, S(R), to be S,(R) = S,(R). A structure theorem is proved for S,(R) which (like the results for minimal ideals and minimal right ideals) is particularly informative if R is purely alternative (free of nuclear ideals): that is, in some sense at the opposite extreme from being associative. We also show that S,(R) annihilates the Smiley radical M(R), thus justifying Baer’s name “antiradical” for S,(R). One result of Section 4 is that if A < R and R is semiprime, then S(A) = A n S(R). I n ec ion 5 we consider when a corresponding result S t

An Ideal Theory for Exterior Differential Equations

The main purpose of this paper is to express the arithmetic notion of genus which appears in E. Cartan's Theory of exterior differential equations [2] in terms of ideal theory. Accordingly Sections 1, 2, and 3 of this paper are devoted to a discussion of ideal theory in Grassmann algebras. Let V be an n dimensional vector space and let V* be its dual space. Let A(V) and A(V*) denote the Grassmann algebra over V and V* respectively. Now A(V) and A(V*) are dually paired to the reals. Let W be a subspace of V and let A(W) be the Grassmann algebra over W. Then A(W) C A(V). We wish to study the annihilator space in A(V*) of A(W). In Theorem 2.3 we show that this is exactly the homogeneous ideal generated in A(V*) by the annihilator of W. An ideal generated by a subspace of V* will be called a one generated ideal. It should be remarked that the integral elements of E. Cartan are precisely Grassmann algebras generated by subspaces of V. We also define a concept of minimal ideal, see Definition 3.1, for a given ideal W and prove that under certain conditions on 9a the dimension of the covector space which generates a minimal ideal is an invariant of the ideal Wf. We call this number the dimension of 2f when it exists. Sections 4, 5, and 6 are devoted to applications of the above techniques to the study of the characteristic vector space, the genus of a differential ideal and the problem of prolongation, respectively. We prove that the characteristic vector space is contained in the intersection of the annihilator spaces of the minimal ideals for the given ideal W. We also find a condition for these two spaces to coincide. Theorem 5.1 shows that for one and two generated ideals Xf, the genus at a point is either one or the dimension of W. In Theorem 5.2 we give a necessary and sufficient condition for the genus to be one. In Section 6 we give an invariant formulation of the process of prolongation and in Theorem 6.2 we obtain a characterization of certain integral ideals in the prolonged system. We will assume in presenting the material of this paper that the reader is familiar with [1] and [2]. The author would like to acknowledge the fact that his interest in this subject stems from a course given by Professor S. S. Chern at the University of Chicago in 1952-1953 and that much of the exposition in paragraphs 5 and 6 follow the ideas in [3].

Topological invariance of ideals in mobs

A mob is a Hausdorff space together with a continuous associative multiplication. In all that follows S will be a compact mob. A set TCS is a left ideal if TpD and if STCT. It is clear how to define right ideal and (two-sided) ideal. Numakura [7] has shown that S contains minimal ideals of all three sorts and these are closed sets. We let K be the minimal ideal of S. Improving some results of [11], we show among other things that, with additional assumptions on S, it is possible to give a completely topological definition of K. It will be seen also that if N is a sufficiently large subgroup of S, then the cohomology structure of S is the same as that of N. This will be done by showing that N= K. From this it follows that N is a homomorphic retract of S. But N need not be a deformation retract of S, see [3 ]. The Alexander-Kolmogoroff cohomology group of the space X with coefficient group G will be denoted by Hn(X, G), Spanier [8]. We sometimes write Hn(X) for Hn(X, G). It is possible to define a dimension function (Haskell Cohen [4]) by letting cd(X, G)<?n if the natural homomorphism Hn(X, G) into Hn(A, G) is onto for each closed A CX. If X is compact Hausdorff then cd(X, integers) is the covering dimension, [1] and [5]. Cohen [4] showed that cd(X, G) cannot exceed the covering dimension for a compact X. If hCHn(X), then hi A will denote the image of h in Hn(A) under the natural homomorphism, A CX. A continuum is a compact connected Hausdorff space.