Extending Jordan Ideals and Jordan Homomorphisms of Symmetric Elements in a Ring with Involution

1972 ◽  
Vol 24 (1) ◽  
pp. 50-59 ◽  
Author(s):  
Kirby C. Smith
Keyword(s):  
Associative Ring ◽  
Background Information ◽  
Matrix Rings ◽  
Jordan Ring ◽  
Jordan Homomorphism ◽  
Rings With Involution ◽  
Jordan Homomorphisms ◽  
Definition Of ◽  
Natural Way

In this work, we show how the ideas in [3, pp. 6-12] can be used to give conditions under which Jordan ideals in the set of symmetric elements in an associative ring R with involution extend to associative ideals of R in a natural way. We also give conditions under which a Jordan homomorphism of the set of symmetric elements will extend to an associative homomorphism of R. Such work has been done on matrix rings with involution in [5; 6]. An abstract definition of a Jordan ring may be found in [3] as well as other background information.

A characterization of Jordan and 5-Jordan homomorphisms between Banach algebras

2018 ◽  
Vol 11 (02) ◽  
pp. 1850021 ◽  
Author(s):  
A. Zivari-Kazempour
Keyword(s):  
Banach Algebra ◽  
Banach Algebras ◽  
Commutative Banach Algebra ◽  
Jordan Homomorphism ◽  
Jordan Homomorphisms ◽  
Unital Banach Algebra

We prove that each surjective Jordan homomorphism from a Banach algebra [Formula: see text] onto a semiprime commutative Banach algebra [Formula: see text] is a homomorphism, and each 5-Jordan homomorphism from a unital Banach algebra [Formula: see text] into a semisimple commutative Banach algebra [Formula: see text] is a 5-homomorphism.

On Strongly Groupoid Graded Rings and the Corresponding Clifford Theorem

2006 ◽  
Vol 13 (02) ◽  
pp. 181-196 ◽  
Author(s):  
Gongxiang Liu ◽  
Fang Li
Keyword(s):  
Main Task ◽  
Graded Rings ◽  
Graded Ring ◽  
Matrix Rings ◽  
Definition Of

In this paper, we introduce the definition of groupoid graded rings. Group graded rings, (skew) groupoid rings, artinian semisimple rings, matrix rings and others can be regarded as special kinds of groupoid graded rings. Our main task is to classify strongly groupoid graded rings by cohomology of groupoids. Some classical results about group graded rings are generalized to groupoid graded rings. In particular, the Clifford Theorem for a strongly groupoid graded ring is given.

scholarly journals Characterization of $n$-Jordan Homomorphisms on Rings

2021 ◽  
Vol 53 ◽  
Author(s):  
Abbas Zivari-kazempour ◽  
Mohammad Valaei
Keyword(s):  
Jordan Homomorphism ◽  
Jordan Homomorphisms

In this paper, we prove that if $\varphi:\mathcal{R}\longrightarrow\mathcal{R}'$ is an $n$-Jordan homomorphism, where $\mathcal{R}$ has a unit $e$, then the map $a\longmapsto \varphi(e)^{n-2}\varphi(a)$ is a Jordan homomorphism.  As a consequence we show, under special hypotheses, that each $n$-Jordan homomorphism is an $n$-homomorphism.

Green Schools as Teaching Tools

2015 ◽  
pp. 155-170
Author(s):  
Charles F. Carrick ◽  
Douglas B. Caywood
Keyword(s):  
Background Information ◽  
Discrete Elements ◽  
Green Schools ◽  
Teaching Activities ◽  
Working Definition ◽  
School Programs ◽  
Subject Areas ◽  
Green School ◽  
Definition Of ◽  
K 12

This chapter is meant to serve as both a resource and as an aid for K-12 teachers who are interested in incorporating the philosophy and various aspects of the green school into their day-to-day teaching activities. A working definition of green schools and a summary of their benefits are presented as background information for teachers unfamiliar with the movement. Suggested steps for greening schools and classrooms are provided for those who may be interested in advancing the concept in their particular situations. Throughout the chapter, the school is highlighted as a laboratory for practicing conservation. To that end, discrete elements of green design are presented as suggested subject areas. Successful green school programs are identified as an additional resource. Finally, suggested green activities for the classroom are provided for interested instructors.

Making Research Methods Instruction Relevant for Prospective Principals

2018 ◽  
pp. 260-283
Author(s):  
Mindy Crain-Dorough ◽  
Adam C. Elder
Keyword(s):  
School Improvement ◽  
Research Methods ◽  
Data Use ◽  
Research Process ◽  
Background Information ◽  
Research On Teaching ◽  
Definition Of ◽  
Taking Action ◽  
Prospective Principals ◽  
Effective Data Use

In this chapter, the authors describe the specific research skills to be developed for prospective principals in preparation for effective data use for school improvement. Relevant background information is provided regarding effective data use leadership, definition of data literacy, standards for principal preparation in data use, research on teaching research methods, and a comparison of the research process and the data-informed decision-making (DIDM) process. These skills are organized and reported in the chapter by steps in the DIDM research process. These steps include goal setting/problem formation, using previous research, planning for data collection, obtaining or collecting data, analyzing data (transforming data into information), and interpreting/taking action/making decisions.

Personalized Learning Plans and Competency-Based Education

2020 ◽  
pp. 28-53
Keyword(s):  
School Personnel ◽  
Personalized Learning ◽  
Background Information ◽  
Competency Based Education ◽  
History Of ◽  
Competency Based ◽  
Postsecondary Plans ◽  
Definition Of ◽  
K 12 ◽  
Learning Plans

This chapter is designed to inform teachers, administrators, and educational policymakers on the background of personalized learning plans (PLPs), the definition of a PLP, components of a PLP, and the research involving the use of PLPs. In recent years, states have spearheaded initiatives involving PLPs, either requiring the use of them through mandates or encouraging school personnel to use some sort of individualized plans for students to connect their K-12 experiences with postsecondary plans. The chapter also addresses competency-based education, which is often implemented in conjunction with personalized learning. The chapter provides an introduction, background information, and a brief history of PLPs and competency-based education. Additional resources are included as well.

Homomorphisms of Rings with Involution

1974 ◽  
Vol 26 (5) ◽  
pp. 1098-1108 ◽  
Author(s):  
Charles Lanski
Keyword(s):  
Matrix Rings ◽  
Rings With Involution

The purpose of this paper is to examine the extent to which a homomorphism of a ring with involution is determined by its action on the symmetric elements of the ring. Assuming that the ring is "suitably free" of 2 × 2 matrix rings, we show that any homomorphism is uniquely determined if its image is semi-prime without nonzero central ideals. To obtain this result we first investigate automorphisms of quotients of rings with involution.

Some Radical Properties of Jordan Matrix Rings

1979 ◽  
Vol 31 (1) ◽  
pp. 189-196
Author(s):  
Michael Rich
Keyword(s):  
Unique Solution ◽  
Identity Element ◽  
Alternative Ring ◽  
Matrix Rings ◽  
Jordan Ring ◽  
Jordan Matrix ◽  
Image Position ◽  
Canonical Involution

Let A be a ring (not necessarily associative) in which 2x = a has a unique solution for each a ∈ A. Then it is known that if A contains an identity element 1 and an involution j : x ↦ x and if Ja is the canonical involution on An determined by where the ai al−l, 1 ≦ i ≦ n are symmetric elements in the nucleus of A then H(An, Ja), the set of symmetric elements of An, for n ≧ 3 is a Jordan ring if and only if either A is associative or n = 3 and A is an alternative ring whose symmetric elements lie in its nucleus [2, p. 127].

scholarly journals Near-rings of quotients of endomorphism near-rings

1975 ◽  
Vol 19 (4) ◽  
pp. 345-352 ◽  
Author(s):  
Michael Holcombe
Keyword(s):  
Additive Group ◽  
Finite Products ◽  
Definition Of ◽  
Group Object ◽  
Rings Of Quotients ◽  
Natural Way

Let be a category with finite products and a final object and let X be any group object in . The set of -morphisms, (X, X) is, in a natural way, a near-ring which we call the endomorphism near-ring of X in Such nearrings have previously been studied in the case where is the category of pointed sets and mappings, (6). Generally speaking, if Γ is an additive group and S is a semigroup of endomorphisms of Γ then a near-ring can be generated naturally by taking all zero preserving mappings of Γ into itself which commute with S (see 1). This type of near-ring is again an endomorphism near-ring, only the category is the category of S-acts and S-morphisms (see (4) for definition of S-act, etc.).

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