Smoothing of a Ring Homomorphism Along a Section

1983 ◽  
pp. 5-31 ◽  
Author(s):  
Michael Artin ◽  
Jan Denef
Keyword(s):  
Ring Homomorphism

scholarly journals Universal mapping properties of some pseudovaluation domains and related quasilocal domains

2006 ◽  
Vol 2006 ◽  
pp. 1-12
Author(s):  
Ahmed Ayache ◽  
David E. Dobbs ◽  
Othman Echi
Keyword(s):  
Positive Integer ◽  
Field Condition ◽  
Ring Homomorphism ◽  
Valuation Domain ◽  
Integral Domains ◽  
Unital Ring ◽  
Local Homomorphism

If(R,M)and(S,N)are quasilocal (commutative integral) domains andf:R→Sis a (unital) ring homomorphism, thenfis said to be astrong local homomorphism(resp.,radical local homomorphism) iff(M)=N(resp.,f(M)⊆Nand for eachx∈N, there exists a positive integertsuch thatxt∈f(M)). It is known that iff:R→Sis a strong local homomorphism whereRis a pseudovaluation domain that is not a field andSis a valuation domain that is not a field, thenffactors via a unique strong local homomorphism through the inclusion mapiRfromRto its canonically associated valuation overring(M:M). Analogues of this result are obtained which delete the conditions thatRandSare not fields, thus obtaining new characterizations of wheniRis integral or radicial. Further analogues are obtained in which the “pseudovaluation domain that is not a field” condition is replaced by the APVDs of Badawi-Houston and the “strong local homomorphism” conditions are replaced by “radical local homomorphism.”

Generalized Casimir Operators

1969 ◽  
Vol 21 ◽  
pp. 1496-1505
Author(s):  
A. J. Douglas
Keyword(s):  
Finite Group ◽  
Casimir Operator ◽  
Identity Element ◽  
Ring Homomorphism ◽  
Fixed Ring ◽  
Injective Modules ◽  
Casimir Operators ◽  
Maschke’S Theorem ◽  
A Finite Group ◽  
Special Case

Throughout this paper, S will be a ring (not necessarily commutative) with an identity element ls ≠ 0s. We shall use R to denote a second ring, and ϕ: S→ R will be a fixed ring homomorphism for which ϕ1S = 1R.In (7), Higman generalized the Casimir operator of classical theory and used his generalization to characterize relatively projective and injective modules. As a special case, he obtained a theorem which contains results of Eckmann (3) and of Higman himself (5), and which also includes Gaschütz's generalization (4) of Maschke's theorem. (For a discussion of some of the developments of Maschke's idea of averaging over a finite group, we refer the reader to (2, Chapter IX).) In the present paper, we define the Casimir operator of a family of S-homomorphisms of one R-module into another, and we again use this operator to characterize relatively projective and injective modules.

On Steinitz-like properties in amalgamated algebras along an ideal

2019 ◽  
Vol 19 (12) ◽  
pp. 2050237
Author(s):  
Mourad El Maalmi ◽  
Hakima Mouanis
Keyword(s):  
Sufficient Conditions ◽  
Ring Homomorphism ◽  
Necessary And Sufficient Conditions ◽  
Necessary And Sufficient

Let [Formula: see text] be a ring homomorphism, [Formula: see text] be an ideal of [Formula: see text] and [Formula: see text] the amalgamation of [Formula: see text] with [Formula: see text] along [Formula: see text] with respect to [Formula: see text]. In this paper, we provide necessary and sufficient conditions for [Formula: see text] to be a Steinitz ring, semi-Steinitz ring, and weakly semi-Steinitz. Then we construct new original examples of weakly semi-Steinitz rings which are not semi-Steinitz rings and semi-Steinitz rings which are not Steinitz rings.

scholarly journals Intuitionistic Fuzzy Normed Subrings and Intuitionistic Fuzzy Normed Ideals

Mathematics ◽  
2020 ◽  
Vol 8 (9) ◽  
pp. 1594
Author(s):  
Nour Abed Alhaleem ◽  
Abd Ghafur Ahmad
Keyword(s):  
Inverse Image ◽  
Ring Homomorphism ◽  
Basic Properties ◽  
Intuitionistic Fuzzy ◽  
Normed Ring

The main goal of this paper is to introduce the notion of intuitionistic fuzzy normed rings and to establish basic properties related to it. We extend normed rings by incorporating the idea of intuitionistic fuzzy to normed rings, we develop a new structure of fuzzy rings which will be called an intuitionistic fuzzy normed ring. As an extension of intuitionistic fuzzy normed rings, we define the concept of intuitionistic fuzzy normed subrings and intuitionistic fuzzy normed ideals. Some essential operations specially subset, complement, union, intersection and several properties relating to the notion of generalized intuitionistic fuzzy normed rings are identified. Homomorphism and isomorphism of intuitionistic fuzzy normed subrings are characterized. We identify the image and the inverse image of intuitionistic fuzzy normed subrings under ring homomorphism and study their elementary properties. Some properties of intuitionistic fuzzy normed rings and relevant examples are presented.

scholarly journals Fuzzy Normed Rings

Symmetry ◽  
2018 ◽  
Vol 10 (10) ◽  
pp. 515 ◽  
Author(s):  
Aykut Emniyet ◽  
Memet Şahin
Keyword(s):  
Prime Ideal ◽  
Fuzzy Sets ◽  
Maximal Ideal ◽  
Ring Homomorphism ◽  
Ring Theory ◽  
Basic Properties ◽  
Normed Ring ◽  
Normed Ideal ◽  
Algebraic Properties ◽  
Definition Of

In this paper, the concept of fuzzy normed ring is introduced and some basic properties related to it are established. Our definition of normed rings on fuzzy sets leads to a new structure, which we call a fuzzy normed ring. We define fuzzy normed ring homomorphism, fuzzy normed subring, fuzzy normed ideal, fuzzy normed prime ideal, and fuzzy normed maximal ideal of a normed ring, respectively. We show some algebraic properties of normed ring theory on fuzzy sets, prove theorems, and give relevant examples.

On Nagata rings with amalgamation properties

2020 ◽  
pp. 2150136
Author(s):  
Ali Molkhasi ◽  
Kar Ping Shum
Keyword(s):  
Commutative Rings ◽  
Ring Homomorphism ◽  
Nagata Ring ◽  
Amalgamated Algebra

Let [Formula: see text] and [Formula: see text] be two commutative rings with unity, let [Formula: see text] be an ideal of [Formula: see text] and [Formula: see text] be a ring homomorphism. In this paper, we give a characterization for the amalgamated algebra [Formula: see text] to be a Nagata ring, a strong S-domain, and a catenarian. Also, we investigate the conditions that the ring of Hurwitz series over [Formula: see text] has a complete comaximal factorization.

Hilbert rings and G(oldman)-rings issued from amalgamated algebras

2018 ◽  
Vol 17 (02) ◽  
pp. 1850023 ◽  
Author(s):  
L. Izelgue ◽  
O. Ouzzaouit
Keyword(s):  
New York ◽  
Commutative Algebra ◽  
Quotient Ring ◽  
Sufficient Conditions ◽  
Ring Homomorphism ◽  
Necessary And Sufficient Conditions ◽  
Local Rings ◽  
Finitely Generated ◽  
Necessary And Sufficient ◽  
The Way

Let [Formula: see text] and [Formula: see text] be two rings, [Formula: see text] an ideal of [Formula: see text] and [Formula: see text] be a ring homomorphism. The ring [Formula: see text] is called the amalgamation of [Formula: see text] with [Formula: see text] along [Formula: see text] with respect to [Formula: see text]. It was proposed by D’anna and Fontana [Amalgamated algebras along an ideal, Commutative Algebra and Applications (W. de Gruyter Publisher, Berlin, 2009), pp. 155–172], as an extension for the Nagata’s idealization, which was originally introduced in [Nagata, Local Rings (Interscience, New York, 1962)]. In this paper, we establish necessary and sufficient conditions under which [Formula: see text], and some related constructions, is either a Hilbert ring, a [Formula: see text]-domain or a [Formula: see text]-ring in the sense of Adams [Rings with a finitely generated total quotient ring, Canad. Math. Bull. 17(1) (1974)]. By the way, we investigate the transfer of the [Formula: see text]-property among pairs of domains sharing an ideal. Our results provide original illustrating examples.

scholarly journals On the Intuitionistic Fuzzy Stability of Ring Homomorphism and Ring Derivation

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
Jaiok Roh ◽  
Ick-Soon Chang
Keyword(s):  
Functional Equation ◽  
Banach Algebra ◽  
Ring Homomorphism ◽  
Normed Algebra ◽  
Intuitionistic Fuzzy ◽  
Jensen Functional Equation ◽  
Jensen Functional ◽  
The Stability ◽  
Ring Derivation

We take into account the stability of ring homomorphism and ring derivation in intuitionistic fuzzy Banach algebra associated with the Jensen functional equation. In addition, we deal with the superstability of functional equationf(xy)=xf(y)+f(x)yin an intuitionistic fuzzy normed algebra with unit.

scholarly journals The Newton complementary dual revisited

2018 ◽  
Vol 17 (01) ◽  
pp. 1850004
Author(s):  
André Dória ◽  
Aron Simis
Keyword(s):  
Ring Homomorphism ◽  
Rational Maps ◽  
Rational Map

This work deals with the notion of Newton complementary duality as raised originally in the work of the second author and B. Costa. A conceptual revision of the main steps of the notion is accomplished which then leads to a vast simplification and improvement of several statements concerning rational maps and their images. A ring-homomorphism like map is introduced that allows for a close comparison between the respective graphs of a rational map and its Newton dual counterpart.

scholarly journals λ-rings, Φ-λ-rings, and Φ-Δ-rings

Filomat ◽  
2019 ◽  
Vol 33 (16) ◽  
pp. 5125-5134
Author(s):  
Rahul Kumar ◽  
Atul Gaur
Keyword(s):  
Prime Ideal ◽  
Commutative Ring ◽  
Ring Homomorphism

Let R be a commutative ring with unity. The notion of ?-rings, ?-?-rings, and ?-?-rings is introduced which generalize the concept of ?-domains and ?-domains. A ring R is said to be a ?-ring if the set of all overrings of R is linearly ordered under inclusion. A ring R ? H is said to be a ?-?-ring if ?(R) is a ?-ring, and a ?-?-ring if ?(R) is a ?-ring, where H is the set of all rings such that Nil(R) is a divided prime ideal of R and ? : T(R) ? RNil(R) is a ring homomorphism defined as ?(x) = x for all x ? T(R). The equivalence of ?-?-rings, ?-?-rings with the latest trending rings in the literature, namely, ?-chained rings and ?-Pr?fer rings is established under some conditions. Using the idealization theory of Nagata, examples are also given to strengthen the concept.

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