This academic article discusses the generalization of p-injective rings and projective modules. It begins with definitions of terms like p-injective modules and epp-rings (rings where every projective module is p-injective). It presents properties of p-injective modules and proves conditions under which a module is p-injective. It also proves properties of epp-rings, including a theorem showing when a ring is a right epp-ring. Examples of epp-rings and rings that are left but not right epp are also given. The article concludes with a proposition about rings that are Morita equivalent.
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In this paper, we prove some common fixed point theorem for weakly compatible maps in intuitionistic fuzzy metric space for two, four and six self mapping.
Bland, Paul E.
Rings and their modules / by Paul E. Bland.
p. cm.(De Gruyter textbook)
Includes bibliographical references and index.
ISBN 978-3-11-025022-0 (alk. paper)
1. Rings (Algebra) 2. Modules (Algebra) I. Title.
QA247.B545 2011
5121.4dc22
2010034731
Equations and Inequalities - Making mathematics accessible to allEduSkills OECD
More than ever, students need to engage with mathematical concepts, think quantitatively and analytically, and communicate using mathematics. All these skills are central to a young person’s preparedness to tackle problems that arise at work and in life beyond the classroom. But the reality is that many students are not familiar with basic mathematics concepts and, at school, only practice routine tasks that do not improve their ability to think quantitatively and solve real-life, complex problems.
How can we break this pattern? This report, based on results from PISA 2012, shows that one way forward is to ensure that all students spend more “engaged” time learning core mathematics concepts and solving challenging mathematics tasks. The opportunity to learn mathematics content – the time students spend learning mathematics topics and practising maths tasks at school – can accurately predict mathematics literacy. Differences in students’ familiarity with mathematics concepts explain a substantial share of performance disparities in PISA between socio-economically advantaged and disadvantaged students. Widening access to mathematics content can raise average levels of achievement and, at the same time, reduce inequalities in education and in society at large.
On essential pseudo principally-injective modulestheijes
Pseudo-injectivity is a generalization of quasi- injectivity. An essential Pseudo-injective module wasintroduced by R. Jaiswal, P.C. Bharadwaj and S. Wongwai [7]. This paper is generalization of above notion with new properties
Abstract: Throughout this paper, all rings have identities and all modules are unitary right modules.
Let R be a ring and M an R-module. A module M is called coretractable if for each proper submodule
N of M, there exists a nonzero homomorphism f from M/N into M. Our concern in this paper is to
develop basic properties of coretractable modules and to look for any relations between coretractable
modules and other classes of modules.
Keyword:. coretractable modules, multiplication modules, quasi-Dedekind modules, rationally–
injective modules, prime module
In this paper , some results about the concept of completely coretractable ring are studied and also another results about
the coretractable module are recalled and investigated.
ABSTRACT--- Let R be a ring with identity and M be an R-module with unitary . In this paper we introduce the
notion of Y-coretractable module . Some basic properties of this class of modules are investigated and some
relationships between these modules and other related concepts are introduced .
Some Remarks on Prime Submodules in Weak Co-Multiplication ModulesIJERA Editor
Throughout this paper, the entire ring will be treated as commutative ring with non-zero identity and all the
modules will be treated as unitary modules. In this paper we shall discuss some remarks on prime submodules in
weak co-multiplication modules.
Some Remarks on Prime Submodules in Weak Co-Multiplication ModulesIJERA Editor
Throughout this paper, the entire ring will be treated as commutative ring with non-zero identity and all the
modules will be treated as unitary modules. In this paper we shall discuss some remarks on prime submodules in
weak co-multiplication modules.
Abstract: Let R be a ring with identity and M be an R-module with unity. In this paper we introduce the notion of C-coretractable modules. Some basic properties of this class of modules are investigated and some relationships between these modules and other related concepts are introduced. Also, we give the notion of strongly C-coretractable and study it comparison with C-coretractable.
Keywords and phrases:(strongly) coretractable modules, (strongly) C-coretractable, mono-coretractable, mono-c-coretractable, simple closed, C-Rickart, multiplication module and quasi-Dedekind module.
The polyadic integer numbers, which form a polyadic ring, are representatives of a fixed congruence class. The basics of polyadic arithmetic are presented: prime polyadic numbers, the polyadic Euler totient function, polyadic division with a remainder, etc. are defined. Secondary congruence classes of polyadic integer numbers, which become ordinary residue classes in the binary limit, and the corresponding finite polyadic rings are introduced. Further, polyadic versions of (prime) finite fields are considered. These can be zeroless, zeroless and nonunital, or have several units; it is even possible for all of their elements to be units. There exist non-isomorphic finite polyadic fields of the same arity shape and order. None of the above situations is possible in the binary case. It is conjectured that any finite polyadic field should contain a certain canonical prime polyadic field as a smallest finite subfield, which can be considered a polyadic analogue of GF (p).
Bland, Paul E.
Rings and their modules / by Paul E. Bland.
p. cm.(De Gruyter textbook)
Includes bibliographical references and index.
ISBN 978-3-11-025022-0 (alk. paper)
1. Rings (Algebra) 2. Modules (Algebra) I. Title.
QA247.B545 2011
5121.4dc22
2010034731
Equations and Inequalities - Making mathematics accessible to allEduSkills OECD
More than ever, students need to engage with mathematical concepts, think quantitatively and analytically, and communicate using mathematics. All these skills are central to a young person’s preparedness to tackle problems that arise at work and in life beyond the classroom. But the reality is that many students are not familiar with basic mathematics concepts and, at school, only practice routine tasks that do not improve their ability to think quantitatively and solve real-life, complex problems.
How can we break this pattern? This report, based on results from PISA 2012, shows that one way forward is to ensure that all students spend more “engaged” time learning core mathematics concepts and solving challenging mathematics tasks. The opportunity to learn mathematics content – the time students spend learning mathematics topics and practising maths tasks at school – can accurately predict mathematics literacy. Differences in students’ familiarity with mathematics concepts explain a substantial share of performance disparities in PISA between socio-economically advantaged and disadvantaged students. Widening access to mathematics content can raise average levels of achievement and, at the same time, reduce inequalities in education and in society at large.
On essential pseudo principally-injective modulestheijes
Pseudo-injectivity is a generalization of quasi- injectivity. An essential Pseudo-injective module wasintroduced by R. Jaiswal, P.C. Bharadwaj and S. Wongwai [7]. This paper is generalization of above notion with new properties
Abstract: Throughout this paper, all rings have identities and all modules are unitary right modules.
Let R be a ring and M an R-module. A module M is called coretractable if for each proper submodule
N of M, there exists a nonzero homomorphism f from M/N into M. Our concern in this paper is to
develop basic properties of coretractable modules and to look for any relations between coretractable
modules and other classes of modules.
Keyword:. coretractable modules, multiplication modules, quasi-Dedekind modules, rationally–
injective modules, prime module
In this paper , some results about the concept of completely coretractable ring are studied and also another results about
the coretractable module are recalled and investigated.
ABSTRACT--- Let R be a ring with identity and M be an R-module with unitary . In this paper we introduce the
notion of Y-coretractable module . Some basic properties of this class of modules are investigated and some
relationships between these modules and other related concepts are introduced .
Some Remarks on Prime Submodules in Weak Co-Multiplication ModulesIJERA Editor
Throughout this paper, the entire ring will be treated as commutative ring with non-zero identity and all the
modules will be treated as unitary modules. In this paper we shall discuss some remarks on prime submodules in
weak co-multiplication modules.
Some Remarks on Prime Submodules in Weak Co-Multiplication ModulesIJERA Editor
Throughout this paper, the entire ring will be treated as commutative ring with non-zero identity and all the
modules will be treated as unitary modules. In this paper we shall discuss some remarks on prime submodules in
weak co-multiplication modules.
Abstract: Let R be a ring with identity and M be an R-module with unity. In this paper we introduce the notion of C-coretractable modules. Some basic properties of this class of modules are investigated and some relationships between these modules and other related concepts are introduced. Also, we give the notion of strongly C-coretractable and study it comparison with C-coretractable.
Keywords and phrases:(strongly) coretractable modules, (strongly) C-coretractable, mono-coretractable, mono-c-coretractable, simple closed, C-Rickart, multiplication module and quasi-Dedekind module.
The polyadic integer numbers, which form a polyadic ring, are representatives of a fixed congruence class. The basics of polyadic arithmetic are presented: prime polyadic numbers, the polyadic Euler totient function, polyadic division with a remainder, etc. are defined. Secondary congruence classes of polyadic integer numbers, which become ordinary residue classes in the binary limit, and the corresponding finite polyadic rings are introduced. Further, polyadic versions of (prime) finite fields are considered. These can be zeroless, zeroless and nonunital, or have several units; it is even possible for all of their elements to be units. There exist non-isomorphic finite polyadic fields of the same arity shape and order. None of the above situations is possible in the binary case. It is conjectured that any finite polyadic field should contain a certain canonical prime polyadic field as a smallest finite subfield, which can be considered a polyadic analogue of GF (p).
Abstract: Here we define a new category which we call “the category of abelian groups over the category of rings” and we denote it by M. This category is the collection of modules over different rings. We have discussed some properties of this category in details. Some results of Ab (the category of abelian groups), Ring (the category of rings) have been proved and used in the proofs of the properties of M in this paper. Also we have defined another new category which we call “the category of semi abelian groups over the category of semi rings” and we have denoted it by M’. We discussed some properties of M’.
Similar to Generalization of p injective rings and projective modules (13)
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Generalization of p injective rings and projective modules
1. Mathematical Theory and Modeling www.iiste.org
ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)
Vol.3, No.6, 2013-Selected from International Conference on Recent Trends in Applied Sciences with Engineering Applications
106
Generalization of p-Injective Rings and Projective Modules
D.S. Singh
Department of Pure & Applied Mathematics
Guru Ghasidas Vishwavidyalaya, Bilaspur (C.G.) INDIA - 495009
Email: vedant_ds@rediffmail.com
Abstract
Any left R-module M is said to be p-injective if for every principal left ideal I of R and any R-homomorphism g:
I→M, there exists y ∈M such that bybg =)( for all b in I. We find that RM is p-injective iff for each r∈R,
x∈M if x∉rM then there exists c∈R with cr = 0 and cx ≠0. A ring R is said to be epp-ring if every projective R-
module is p-injective. Any ring R is right epp-ring iff the trace of projective right R-module on itself is p-
injective. A left epp-ring which is not right epp-ring has been constructed.
Key words: P-injective, epp-ring, f-injective, Artinian, Noetherian.
Subject Classification code: 16D40, 16D50, 16P20.
Introduction: Any ring R is said to be QF if every injective left R-module is projective. Villamayor [3]
characterises a ring R over which every simple R-module is injective Faith call it V-ring. R.R Colby defines a
ring R as a left (right) IF-ring if every injective left (right) R-module is flat. R.Y.C. Ming[4] characterises the
rings over which every simple left R-module is p-injective. Motivated by these ideas here we define a ring R as a
left epp-ring if every left projective R-module is p-injective. Through out, R denotes an associative ring with
identity and R-modules are unitary.
Definition: 1. A left R-module M is called p-injective if for any principal left ideal I of R and any left R-
homomorphism g : I→ M, there exist y ∈ M such that g (b) = by for all b in I.
Definition : 2. A left R-module M is f -injective if, for any finitely generated left ideal I of R and any left R-
homomorphism g:I →M , there exists y∈ M such that g (b) = b y for all b in I.
Proposition: 3(i) Direct product of p-injective modules is p-injective if and only if each factor is p-injective.
(ii) Direct sum of p-injective modules is p-injective if and only if each summand is p-injective. [4]
Proof :-( i) Let the direct product Π
A
Mi be p-injective module, to show each Mi is p-injective. For this consider
the homomorphism fi : I → Mi where I = (a) be any principal left ideal of R generated by 'a'. If ji : Mi → Π
A
Mi
be the inclusion homomorphism then ji o fi : I → Π
A
Mi is a homomorphism. Since Π
A
Mi is p-injective, there
exists (mi) i∈A∈Π Mi such that ji o fi (x) = x (mi)i∈A for all x∈ I.
Let qi : Π
A
Mi → Mi be the projection homomorphism.
Then qi o ji o fi = fi and qi o ji o fi(a) = qi(a(mi)i∈A) = a mi = fi (a). Hence Mi is p-injective.
Conversely let each Mi is p-injective and f:I→ Π
A
Mi be any R-homomorphism. Consider the
homomorphism qi o f: I→Mi. By p-injectivity of Mi there exists mi for each i ∈ A such that qi o f (a) = a mi.
Thus f: I → Π
A
Mi is given by f (a) = a (mi)i∈A. Hence Π
A
Mi is p-injective.
(ii) Its proof is very much similar to (i).
Corollary: 4. Direct sum of injective module is p-injective.
Theorem: 5. Following conditions are equivalent for a left R-module M;
(i) M is left p-injective module.
(ii) For each r∈R, x∈M of x∉rM then there exists c∈R with cr = 0 and cx ≠ 0.
Proof: (i) ⇒ (ii) Suppose (ii) is not true that is for each r∈R, x∈M if x∉rM for some c, cr = 0 and cx = 0. Put I =
(r) = Rr then there is an R-homomorphism f: I→M defined by f(r) = x. By p-injectivity of RM there exists x'∈M
such that f(r) = rx' for all r∈I.
Therefore x = f(r) = rx'∈ rm which is a contradiction to the fact that x∉rM.
(ii) ⇒ (i) Let I = (r) = Rr be any principal left ideal and f : I → M be any R-homomorphism. Suppose f(r) ≠ ry
for some y∈M and cr = 0. This implies that f(r)∉rm and f(cr) = cf(r) = 0 which is a contradiction to the fact that
cr=0 and cx ≠ 0.
Definition: 6. A ring R is right (left) epp-ring if every right (left) projective R-module is p-injective. A ring R is
epp-ring if it is both right as well as left epp-ring.
The trace of an R-module M on M is denoted by
2. Mathematical Theory and Modeling www.iiste.org
ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)
Vol.3, No.6, 2013-Selected from International Conference on Recent Trends in Applied Sciences with Engineering Applications
107
TM(M) = {Imf | f ∈S = EndRM}.
Lemma: 7. [5] (i) Let A ∈ |MR. The map ϕ'(A) : Hom (M, A) ⊗ SM → TM (A) is an isomorphism if and only if
M generates all kernels of homomorphism Mn
→ A, n∈N.
(ii) The left S-module SM is flat if and only if generates all kernels of homomorphism Mn
→ M, n∈N.
Theorem: 8. Following conditions are equivalent for a ring R;
(i) R is right epp-ring.
(ii) The trace of projective module MR on itself is p-injective.
(iii) The trace of right free R-module on itself is p-injective.
Proof: (i) ⇒ (ii) Let R be any right epp-ring i.e. every right projective R-module is p-injective and M be any
projective right R-module. Using the fact that every projective is flat and the tensor product of two flat modules
is flat we get SM ⊗RR ≅ SM is flat. Therefore by Lemma 7 M generates all kernels of the homomorphisms
Mn
→M, n∈N,which implies that ϕ'(M):HomR(M,M)⊗M→ TM(M) is an isomorphism by Lemma 7 that is S ⊗
MR is an isomorphic to TM(M) or MR is isomorphic to TM(M) and so TM(M) is p-injective as R is epp-ring.
(ii) ⇒ (iii) obvious
(iii) ⇒ (i) Assume that for every free right R-module M, TM(M) is p-injective. M would be flat too (free ⇒
projective ⇒ flat). Using the same arguments as in (i) ⇒ (ii) we get TM(M) is isomorphic to M therefore M is p-
injective. Let K be any projective R-module than K would be direct summand of some free R-module say
M but M is p-injective. Since direct summand of p-injective is p-injective [proposition 3(ii)] hence K is p-
injective that is R is a epp-ring.
This theorem is true for a left epp-ring and left projective R-module M too.
Example: 9 [2, R.R. Colby Example 1].
A commutative epp-ring which is not a epp-ring modulo its radical. Let R =Z ⊕
Q
Z
with multiplication
defined by (n1,q1) (n2,q2) = (n1n2, n1q2 + n2q1), ni ∈Z, qi ∈ Q. Then R is a commutative coherent ring with Jacobson
radical
J(R) =
( )n Q
Z
,
,
q
n
=
=
0 0
It is obvious that each finitely generated ideal is principal. Thus R is epp-ring but
R
J(R)
≅ Z which is not a epp-
ring as the homomorphism f: Z → Z defined by f (nx) = x can not be extended to a homomorphism from Z → Z.
Example 10: A left epp-ring which is not a right epp-ring [2, Example 2].
Let R be an algebra over a field F with basis {1, e0, e1, ....x1, x2, .....} for all i, j
ei ej = δi, j ej
xiej = δi, j+1 xi
ei xj = δi j xj
xixj = 0
It can be easily verified that R is left coherent and every R-homomorphism f: RI → RR extends
from RR → RR. Thus RR is p-injective that is R is left epp-ring. However R is not right epp-ring, since the
homomorphism x1R → e0R via x1r→ e0r can not be extended over R.
Preposition: 11. Let R and S are Marita equivalent ring then R is epp-ring if and only if S is epp-ring.
Proof: Let F: R|M → S|M and G: S|M → R|M are category equivalences where R|M and S|M denote the categories
of left R-module and left S-module respectively. Since projectivity is a categorical property we have to show
only that M is p-injective in R|M if and only if F (M) is p-injective in S|M. The sequence with principal ideal I,
0→ I → R →
R
I
→ 0
is exact in S|M if and only if
0→ G (I) → G(R) → G (
R
I
) → 0
is exact in R|M [1, page 224]. And the sequence
3. Mathematical Theory and Modeling www.iiste.org
ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)
Vol.3, No.6, 2013-Selected from International Conference on Recent Trends in Applied Sciences with Engineering Applications
108
0→ Hom R (G (
R
I
), M) → Hom R (G (R), M) → Hom R (G (I), M) → 0
is exact in R|M if and only if
0→ HomS (
R
I
, F(M) ) → HomS (R, F( M) ) → HomS (I, F( M) ) → 0
is an exact sequence in S|M.
That is if M is p-injective in R|M if and only if F(M) is p-injective in S|M.
REFERENCES
1. F.W. Anderson and K.R. Fuller, Rings and categories of Modules, Springer-Verlag, New York,
Heidelberg, 1974.
2. R.R. Colby Rings which have flat Injective Modules, J. Algebra 35 (1975), 239-252.
3. G.O. Michler and O.E. Villamayor, On Rings Whose Simple Modules are Injective, J. Algebra 25
(1973), 185-201.
4. R.Y.C. Ming, On V-rings and Prime Rings, J. Algebra 62 (1980), 13-20.
5. B. Zimmermann-Huisgen, Endomorphism Rings of Self Generators, Pacific J. Math. 61(1) (1975), 587-
602.
4. This academic article was published by The International Institute for Science,
Technology and Education (IISTE). The IISTE is a pioneer in the Open Access
Publishing service based in the U.S. and Europe. The aim of the institute is
Accelerating Global Knowledge Sharing.
More information about the publisher can be found in the IISTE’s homepage:
http://www.iiste.org
CALL FOR PAPERS
The IISTE is currently hosting more than 30 peer-reviewed academic journals and
collaborating with academic institutions around the world. There’s no deadline for
submission. Prospective authors of IISTE journals can find the submission
instruction on the following page: http://www.iiste.org/Journals/
The IISTE editorial team promises to the review and publish all the qualified
submissions in a fast manner. All the journals articles are available online to the
readers all over the world without financial, legal, or technical barriers other than
those inseparable from gaining access to the internet itself. Printed version of the
journals is also available upon request of readers and authors.
IISTE Knowledge Sharing Partners
EBSCO, Index Copernicus, Ulrich's Periodicals Directory, JournalTOCS, PKP Open
Archives Harvester, Bielefeld Academic Search Engine, Elektronische
Zeitschriftenbibliothek EZB, Open J-Gate, OCLC WorldCat, Universe Digtial
Library , NewJour, Google Scholar