scholarly journals Left self distributive near-rings

1990 ◽  
Vol 49 (2) ◽  
pp. 273-296 ◽  
Author(s):  
Gary Birenmeier ◽  
Henry Heatherly
Keyword(s):  
Subdirectly Irreducible ◽  
Left Multiplication ◽  
Nilpotent Elements ◽  
Maximal Ideals ◽  
Ring Endomorphism

AbstractThis paper considers (left) near-rings which satisfy the left self distributive (LSD) identity: abc = abac. This is exactly the class of near-rings for which each left multiplication mapping, τa: x → ax, is a near-ring endomorphism. Simple and subdirectly irreducible ones are classified and semidirect sum decompositions into reduced and nilpotent pieces are given. LSD near-rings with restrictive conditions on nilpotent elements or annihilating sets are considered. Type 1 prime (semiprime) ideals in an LSD near-ring are completely prime (semiprime). Further results on prime and maximal ideals are given. Numerous examples are given to illuminate the theory and to illustrate its limitations. Some analogous theory for right self distributive near-rings is given (those satisfying the identity: abc = acbc).

scholarly journals Rings whose additive endomorphisms are ring endomorphisms

1990 ◽  
Vol 42 (1) ◽  
pp. 145-152 ◽  
Author(s):  
Gary Birkenmeier ◽  
Henry Heatherly
Keyword(s):  
Subdirectly Irreducible ◽  
Finiteness Conditions ◽  
Chain Conditions ◽  
Open Questions ◽  
Open Case ◽  
Ring Endomorphism ◽  
Substantial Progress ◽  
Additive Endomorphism ◽  
Made In

A ring R is said to be an AE-ring if every additive endomorphism is a ring endomorphism. In this paper further steps are made toward solving Sullivan's Problem of characterising these rings. The classification of AE-rings with. R3 ≠ 0 is completed. Complete characterisations are given for AE-rings which are either: (i) subdirectly irreducible, (ii) algebras over fields, or (iii) additively indecomposable. Substantial progress is made in classifying AE-rings which are mixed – the last open case – by imposing various finiteness conditions (chain conditions on special ideals, height restricting conditions). Several open questions are posed.

scholarly journals Simple surjective algebras having no proper subalgebras

1990 ◽  
Vol 48 (3) ◽  
pp. 434-454 ◽  
Author(s):  
Ágnes Szendrei
Keyword(s):  
Subdirectly Irreducible ◽  
Irreducible Algebra ◽  
Minimal Variety ◽  
Locally Finite ◽  
Subdirectly Irreducible Algebra ◽  
Matrix Power

AbstractWe prove that every finite, simple, surjective algebra having no proper subalgebras is either quasiprimal or affine or isomorphic to an algebra term equivalent to a matrix power of a unary permutational algebra. Consequently, it generates a minimal variety if and only if it is quasiprimal. We show also that a locally finite, minimal variety omitting type 1 is minimal as a quasivariety if and only if it has a unique subdirectly irreducible algebra.

Subdirectly Irreducible Semirings and Semigroups Without Nonzero Nilpotents

1973 ◽  
Vol 16 (1) ◽  
pp. 45-47 ◽  
Author(s):  
William H. Cornish
Keyword(s):  
Distributive Lattice ◽  
Subdirectly Irreducible ◽  
Nilpotent Elements ◽  
Zero Divisors

It follows from [1, p. 377, Lemma 1] that a noncommutative subdirectly irreducible ring, with no nonzero nilpotent elements, cannot possess any proper zero-divisors. From [2, p. 193, Corollary 1] a subdirectly irreducible distributive lattice, with more than one element, is isomorphic to the chain with two elements. Hence we can say that a subdirectly irreducible distributive lattice with 0 possesses no proper zero-divisors.

SOME PROPERTIES OF FINITELY DECIDABLE VARIETIES

1992 ◽  
Vol 02 (01) ◽  
pp. 89-101 ◽  
Author(s):  
MATTHEW A. VALERIOTE ◽  
ROSS WILLARD
Keyword(s):  
Order Theory ◽  
Subdirectly Irreducible ◽  
Locally Finite ◽  
First Order ◽  
Minimal Sets ◽  
First Order Theory ◽  
Finite Member ◽  
Transfer Principles

Let [Formula: see text] be a variety whose class of finite members has a decidable first-order theory. We prove that each finite member A of [Formula: see text] satisfies the (3, 1) and (3, 2) transfer principles, and that the minimal sets of prime quotients of type 2 or 3 in A must have empty tails. The first result has already been used by J. Jeong [9] in characterizing the finite subdirectly irreducible members of [Formula: see text] with nonabelian monolith. The second result implies that if [Formula: see text] is also locally finite and omits type 1, then [Formula: see text] is congruence modular.

On a Class of Constacyclic Codes of Length 4ps over Fpm[u]〈 ua 〉

2019 ◽  
Vol 26 (02) ◽  
pp. 181-194 ◽  
Author(s):  
Hai Q. Dinh ◽  
Bac T. Nguyen ◽  
Songsak Sriboonchitta
Keyword(s):  
Principal Ideal ◽  
Constacyclic Codes ◽  
Constacyclic Code ◽  
Chain Ring ◽  
Principal Ideal Ring ◽  
Direct Sum ◽  
Ideal Ring ◽  
Maximal Ideals

For any odd prime p such that pm ≡ 3 (mod 4), consider all units Λ of the finite commutative chain ring [Formula: see text] that have the form Λ = Λ0 + uΛ1 + ⋯ + ua−1 Λa−1, where Λ0, Λ1, …, Λa−1 ∊ 𝔽pm, Λ0 ≠ 0, Λ1 ≠ 0. The class of Λ-constacyclic codes of length 4ps over ℛa is investigated. If the unit Λ is a square, each Λ-constacyclic code of length 4ps is expressed as a direct sum of a −λ-constacyclic code and a λ-constacyclic code of length 2ps. In the main case that the unit Λ is not a square, we prove that the polynomial x4 − λ0 can be decomposed as a product of two quadratic irreducible and monic coprime factors, where [Formula: see text]. From this, the ambient ring [Formula: see text] is proven to be a principal ideal ring, whose maximal ideals are ⟨x2 + 2ηx + 2η2⟩ and ⟨x2 − 2ηx + 2η2⟩, where λ0 = −4η4. We also give the unique self-dual Type 1 Λ-constacyclic codes of length 4ps over ℛa. Furthermore, conditions for a Type 1 Λ-constacyclic code to be self-orthogonal and dual-containing are provided.

scholarly journals On Duo Rings

1960 ◽  
Vol 3 (2) ◽  
pp. 167-172 ◽  
Author(s):  
G. Thierrin
Keyword(s):  
Commutative Rings ◽  
Prime Ideals ◽  
Subdirectly Irreducible ◽  
Nilpotent Elements

Following E. H. Feller [l], a ring R is called a duo ring if every one-sided ideal of R is a two-sided ideal.In the first part of this paper, we give some properties of duo rings and we show that the set of the nilpotent elements of a duo ring R is an ideal, the intersection of the completely prime ideals of R.It is easy to see that every duo ring is a subdirect sum of subdirectly irreducible duo rings. We give in the second part of this paper a characterization of the subdirectly irreducible duo rings. This characterization is quite similar to the characterization of the subdirectly irreducible commutative rings, due to N. H. McCoy [2], whose methods we use.

scholarly journals A note on subdirectly irreducible rings

1984 ◽  
Vol 30 (1) ◽  
pp. 137-141 ◽  
Author(s):  
Shalom Feigelstock
Keyword(s):  
Minimal Ideal ◽  
Subdirectly Irreducible ◽  
Nilpotent Elements

Let R be a commutative subdirectly irreducible ring, with minimal ideal M. It is shown that either R is a field, or M2 = 0. A construction is given which yields commutative sub-directly irreducible rings possessing nonzero-divisors, and nonzero nilpotent elements either with a unity element, or without. Such a ring without unity has been constructed by Divinsky. The same technique enables the construction of subdirectly irreducible rings with mixed additive groups.

Crystalloid Inclusions in Hepatocyte Mitochondria and Their Similarity to Cytoplasmic Crystalloids

1974 ◽  
Vol 32 ◽  
pp. 198-199
Author(s):  
Odell T. Minick ◽  
Hidejiro Yokoo
Keyword(s):  
Liver Disease ◽  
Alcoholic Liver Disease ◽  
Special Interest ◽  
Structural Variations ◽  
Mitochondrial Alterations ◽  
The Matrix ◽  
Crystalloid Inclusions ◽  
Liver Biopsies

Mitochondrial alterations were studied in 25 liver biopsies from patients with alcoholic liver disease. Of special interest were the morphologic resemblance of certain fine structural variations in mitochondria and crystalloid inclusions. Four types of alterations within mitochondria were found that seemed to relate to cytoplasmic crystalloids.Type 1 alteration consisted of localized groups of cristae, usually oriented in the long direction of the organelle (Fig. 1A). In this plane they appeared serrated at the periphery with blind endings in the matrix. Other sections revealed a system of equally-spaced diagonal lines lengthwise in the mitochondrion with cristae protruding from both ends (Fig. 1B). Profiles of this inclusion were not unlike tangential cuts of a crystalloid structure frequently seen in enlarged mitochondria described below.

Evidence for a shear mechanism for the precipitation of γ-zirconium hydride in zirconium

1978 ◽  
Vol 36 (1) ◽  
pp. 658-659
Author(s):  
G.J.C. Carpenter
Keyword(s):  
Elevated Temperature ◽  
Strain Field ◽  
Zirconium Hydride ◽  
Zirconium Atom ◽  
Atom Diffusion ◽  
Solvus Temperature ◽  
Hexagonal Close Packed ◽  
Partial Dislocations ◽  
The Face

In zirconium-hydrogen alloys, rapid cooling from an elevated temperature causes precipitation of the face-centred tetragonal (fct) phase, γZrH, in the form of needles, parallel to the close-packed <1120>zr directions (1). With low hydrogen concentrations, the hydride solvus is sufficiently low that zirconium atom diffusion cannot occur. For example, with 6 μg/g hydrogen, the solvus temperature is approximately 370 K (2), at which only the hydrogen diffuses readily. Shears are therefore necessary to produce the crystallographic transformation from hexagonal close-packed (hep) zirconium to fct hydride.The simplest mechanism for the transformation is the passage of Shockley partial dislocations having Burgers vectors (b) of the type 1/3<0110> on every second (0001)Zr plane. If the partial dislocations are in the form of loops with the same b, the crosssection of a hydride precipitate will be as shown in fig.1. A consequence of this type of transformation is that a cumulative shear, S, is produced that leads to a strain field in the surrounding zirconium matrix, as illustrated in fig.2a.

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