The aim of this paper is to prove some concepts related with strong regularity and strong reducedness in near rings. Fully idempotent nearrings are defined and characterized which yields information on the lattice of ideals of fully idempotent rings and nearrings. In this paper, we have proved some more results on right bipotent near rings by using the concepts of s near ring. Specifically, if k is a field and x is an indeterminate, then the ring of formal power series kx is a regular local ring having krull dimension 1. The concept of noetherian nearrings and noetherian dnear rings was. Also, some properties of p idempotents, p centers, p identities in p completely prime near rings are investigated. We give some characteriza tions of s weakly regular nearripgs. We arrive a conclusion that all reduced and regular near rings are strongly regular and strongly reduced near rings. In this paper the terms, regular nearrings,r regular nearrings, symmetric near ring, weakly regular nearring, completely prime ideal, 1prime. On the other hand, if dimrhxi dimr, then we may choose a chain of primes q 0 q 1 q d r which remains a proper sequence of primes after passing to rhxi. In this paper the terms, regular nearrings,r regular nearrings, symmetric nearring, weakly regular nearring, completely prime ideal, 1prime. Right regularity and right strong regularity are defined in a symmetric way and the definition of regularity shall be the same as for rings. So the coming theorems present several representations of elements of quasiideals of a pregular nearring.
On the other hand, if dimrhxi dimr, then we may choose a chain of primes q 0 q 1 q d r which remains a proper sequence of. We now have the following inclusions of various types of rings. We arrive a conclusion that all reduced and regular nearrings are strongly regular and strongly reduced nearrings. Let eo denote the dimension of the kvector space m2. Strongly regular nearrings cambridge university press. Pdf note on strongly regular nearrings researchgate. P regular and p strongly regular nearrings are studied. W e also prove the c haracterization of strongly regular. We investigated some basic properties for rregular nearrings. According to mason 1 a right nearring n is called i left right strongly regular if for every a there is an x in n such that a xa2 a a2x and ii.
For the only if part, we observe that when a 0, clearly a. Regular local rings i university of california, berkeley. Minimal ideals and primitivity in near rings wendt, gerhard, taiwanese journal of mathematics, 2019. A near ring n is said to be m regular near ring if for each a n, where exists an element b n such that a abma where m. Representation of near ring morita contexts and recognizing morita near rings shoji, kyuno and stefan, veldsman, tsukuba journal of mathematics, 1996. Introduction to groups, rings and fields ht and tt 2011 h. Journal of algebra 26, 440445 1973 regular local rings and excellent rings paolo valabrega instituto matematico, universita di torino, turin, italy received march 7, 1972 introduction it is well known that every regular local ring of dimension 1, whose field of fractions has characteristic 0, is excellent the statement is true for dedekind domains. Get a printable copy pdf file of the complete article 684k, or click on a page image below to browse page by page. Regular local rings i april, 2016 let obe a noetherian local ring with maximal ideal m and residue eld k. Proceedings of the edinburgh mathematical society, vol. Dec 01, 2010 we observe that, when n is a commutative near ring, the concepts of strong s. Pdf on oct 1, 1986, motoshi hongan and others published note on strongly regular nearrings find, read and cite all the research you need on.
In this paper, we show some characteristics of quasiideals of a pregular nearring and in particular, we consider the representations of elements of quasiideals of a pregular nearring related with the ideal p. We shall add to this body of results several commutativity theorems for nearrings admitting suitablyconstrained derivations. The concept of noetherian nearrings and noetherian d. Nearrings in which each element is a power of itself volume 2 issue 3 howard e. Pdf note on strongly regular nearrings motoshi hongan. Local rings cm rings gorenstein rings complete intersections rlrs 1. Pproperties in nearrings atagun journal of mathematical. Weakly and strongly regular nearrings algebra colloquium. So the coming theorems present several representations of elements of quasiideals of a p regular near ring. Also we discuss the idea of semicentral in near rings. The aim of this paper is to prove some concepts related with strong regularity and strong reducedness in nearrings. This is a much higher resolution than can be obtained in a conventional photo of the rings. The quasiradical and the radicalsubgroup of a regular near ring are shown to be 0. The space of prime ideals is topologized and a sheaf representation is given for a class of fully idempotent nearrings.
We have shown that a zerosymmetric nearring n is pstrongly. In this paper, we prove some basic properties of left weakly regular nearrings. If a is regular then so is ax, with dimension one greater than that of a. If you do know that local regular rings are integrally closed, you know that a regular ring has all its localizations at prime ideals integrally closed. Using this characterization we deduce several characterizations of regular nearrings theorem 2. P completely prime ideals are introduced and some characterizations of completely prime near rings are provided.
However, formatting rules can vary widely between applications and fields of interest or study. Using the occultations of stars by the rings of saturn, astronomers have been able to measure details in the ring structure to a resolution of 10 km. Therefore, the only piece of knowledge you need to reach the conclusion you want is that. For a little extra, we invite you to upgrade your side, beverage or both. In this study, the ifp condition in a near ring is extended to the ideals in near rings. But again, we can assume that q 0 p i for some i since q 0 must be some minimal prime. If a regular ring is noetherian or perfect left or right, then it is a classical semisimple ring. Ege university, department of mathematics, science faculty. P regular and p strongly regular near rings are studied. On strongly regular nearrings cambridge university press. Fully idempotent nearrings and sheaf representations. A fact about regular local rings university of utah. Examples of regular rings include fields of dimension zero and dedekind domains. Using this characterization we deduce several characterizations of regular near rings theorem 2.
Now, in the present paper we introduce and study a new generalization of right resp. Nearrings in which each element is a power of itself. Regular local rings and excellent rings sciencedirect. Regular duo elements of abstract affine near rings yakabe, iwao, proceedings of the japan academy, series a, mathematical sciences, 1990. Reliable information about the coronavirus covid19 is available from the world health organization current situation, international travel. We furnish below a characterisation of regular nearrings. Oclcs webjunction has pulled together information and resources to assist library staff as they consider how to handle coronavirus. On s near rings and s near rings with right bipotency. P completely prime ideals are introduced and some characterizations of completely prime nearrings are provided. We give an affirmative answer to the question whether a left weakly regular nearring with left unity and satisfying the ifp is also right weakly regular. Several mathematicians studied and developed various types of near rings such as boolean near rings, ifp near ring, left bipotent near ring, p abelian and strongly regular near ring and strong ifp near rings. Regular duo elements of abstract affine nearrings yakabe, iwao, proceedings of the japan academy, series a, mathematical sciences, 1990. In this paper, we show some characteristics of quasiideals of a p regular near ring and in particular, we consider the representations of elements of quasiideals of a p regular near ring related with the ideal p. Then i abm,bma are idempotents ii abmn an and nbma na proof i let a n.
We have shown that a reduced nearring n is left wweakly regular if and only if nip is a simple domain for every prime ideal p of. We also determine a characterization of strictly semisimple near rings among near rings with no nonzero nilpotent elements theorem 2. We give some characteriza tions of sweakly regular nearripgs. Sorry, we are unable to provide the full text but you may find it at the following locations. We characterize these modules in terms of certain restricted injectivity properties proposition 2. Recommended problem, partly to present further examples or to extend theory. Many problems, even some that seem \global, can be attacked by. Regular ring plays an important role in structure theory of rings which was first introduced by vonneumann. In this paper the terms, regular nearrings,rregular nearrings, symmetric nearring, weakly regular nearring, completely prime ideal, 1prime ideal, 1semiprime ideals are introduced. We give an affirmative answer to the question whether a left weakly regular near ring with left unity and satisfying the ifp is also right weakly regular. Notes on prime nearrings with multiplicative derivation 356 in 7, herstein has proved that if r is a prime ring of characteristic different from 2 and if d is a nonzero derivation of r such that dr z, then r is commutative. In some of the literature a normal ring is also required to be an integral domain. We investigated some basic properties for r regular near rings. An ideal in a regular ring is a regular ring possibly without unit element.
Regular local rings let a be a noetherian local ring, with maximal ideal m and residue eld k. If n is a right module over n itself, we note that assnn ann 6y of goodearly,and war. Regular nearrings without nonzero nilpotent elements. We shall add to this body of results several commutativity theorems for near rings admitting suitablyconstrained derivations. Throughout this paper, by a nearrings we mean zero symmetric nearring for the basic terminology and notation the reader referred to gunter pilz 3. Throught this paper by a nearring we mean a zerosymmetric right near ring. A ring of matrices over a regular ring is again a regular ring.
Each section is followed by a series of problems, partly to check understanding marked with the letter \r. We also determine a characterization of strictly semisimple nearrings among nearrings with no nonzero nilpotent elements theorem 2. While in mccoys paper, most attention was given to. Strongly semi prime noetherian regular delta near rings. Buy toc in rings regular desktop font from ingrimayne type on. The structure theory of complete local rings introduction in the study of commutative noetherian rings, localization at a prime followed by completion at the resulting maximal ideal is a way of life.
We have shown that a reduced nearring n is left w weakly regular if and only if nip is a simple domain for every prime ideal p of. A near ring n is defined to be right bipotent if ana2 n for each a in n. The class of regular rings is closed under the formation of direct products and quotient rings. A near ring n is called an ifp near ring provided that for all a, b, n. Any discrete valuation ring is a regular local ring of dimension 1 and the regular local rings of dimension 1 are exactly the discrete valuation rings. Full text full text is available as a scanned copy of the original print version.
Onions, enriched bleached wheat flour wheat flour, niacin, ferrous sulfate, thiamine mononitrate, riboflavin, folic acid, vegetable oil soybean andor canola, beer water, malted barley, corn syrup, hops, salt, yellow corn flour. In particular if k is a field, the polynomial ring, is regular. Choose any butterburger or other favorite, pick the perfect classic side and a regular drinkand save. Notes on prime nearrings with multiplicative derivation. In 3, bell and kappe have proved that d is a derivation of r which is either a homomorphism or an antihomomorphism in. The literature on nearrings contains a number of theorems asserting that certain conditions implying commutativity in rings imply multiplicative or additive commutativity in special classes of nearrings. In this paper the terms, regular near rings,r regular near rings, symmetric near ring, weakly regular near ring, completely prime ideal, 1prime ideal, 1semiprime ideals are introduced. The theory and its applications, volume 23 1st edition. Any localization of a regular ring is regular as well. Analogues of ring theory results concerning the jacobson radical of a regular ring are obtained for near rings with a twosided zero. Also, some properties of p idempotents, p centers, p identities in p completely prime nearrings are investigated.
Available formats pdf please select a format to send. Regular local rings university of california, berkeley. Numerous and frequentlyupdated resource results are available from this search. In this paper, we generalize further and introduce tile notion of weakly b regular nearrings and obtain a characterization of tile same. Also we discuss the idea of semicentral in nearrings. In this paper, we prove some basic properties of left weakly regular near rings. Pdf in this paper we introduce the notion of pstrongly regular nearring. The structure theory of complete local rings introduction.
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