Injective endomorphism modules

2011 ◽  
pp. 249-264
Author(s):  
Theodore G. Faticoni
Keyword(s):  
Injective Endomorphism

scholarly journals Irreducible skew polynomials over domains

2021 ◽  
Vol 29 (3) ◽  
pp. 75-89
Author(s):  
C. Brown ◽  
S. Pumplün
Keyword(s):  
Polynomial Ring ◽  
Skew Polynomial Ring ◽  
Weyl Algebras ◽  
Skew Polynomials ◽  
Low Degree ◽  
Skew Polynomial ◽  
Low Degree Polynomials ◽  
Injective Endomorphism

Abstract Let S be a domain and R = S[t; σ, δ] a skew polynomial ring, where σ is an injective endomorphism of S and δ a left σ -derivation. We give criteria for skew polynomials f ∈ R of degree less or equal to four to be irreducible. We apply them to low degree polynomials in quantized Weyl algebras and the quantum planes. We also consider f(t) = tm − a ∈ R.

scholarly journals ORE EXTENSIONS SATISFYING A POLYNOMIAL IDENTITY

2006 ◽  
Vol 05 (03) ◽  
pp. 287-306 ◽  
Author(s):  
ANDRÉ LEROY ◽  
JERZY MATCZUK
Keyword(s):  
Polynomial Identity ◽  
Semiprime Ring ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Ore Extension ◽  
Ore Extensions ◽  
Necessary And Sufficient ◽  
Coefficient Ring ◽  
Injective Endomorphism

Necessary and sufficient conditions for an Ore extension S = R[x;σ,δ] to be a PI ring are given in the case σ is an injective endomorphism of a semiprime ring R satisfying the ACC on annihilators. Also, for an arbitrary endomorphism τ of R, a characterization of Ore extensions R[x;τ] which are PI rings is given, provided the coefficient ring R is noetherian.

The Preimage of a Coordinate

2010 ◽  
Vol 17 (01) ◽  
pp. 43-46
Author(s):  
Sheng-Jun Gong ◽  
Jie-Tai Yu
Keyword(s):  
Characteristic Zero ◽  
By Products ◽  
Injective Endomorphism

Let K be a field of characteristic zero. Based on the degree estimate of Makar-Limanov and Yu, we prove that the preimage of a coordinate under an injective endomorphism of K〈x, y〉 is also a coordinate. As by-products, we give new proofs of the following results: (1) the preimage of a coordinate under an injective endomorphism of K[x,y] is also a coordinate; (2) any automorphism of K[x,y] or K〈x, y〉 is tame.

scholarly journals Spectral triples on irreversible C∗-dynamical systems

2021 ◽  
Author(s):  
Valeriano Aiello ◽  
Daniele Guido ◽  
Tommaso Isola
Keyword(s):  
Dynamical Systems ◽  
Product Formula ◽  
Crossed Product ◽  
Spectral Triple ◽  
Crossed Products ◽  
Spectral Triples ◽  
Main Assumption ◽  
Noncommutative Spaces ◽  
Covering Projection ◽  
Injective Endomorphism

Given a spectral triple on a [Formula: see text]-algebra [Formula: see text] together with a unital injective endomorphism [Formula: see text], the problem of defining a suitable crossed product [Formula: see text]-algebra endowed with a spectral triple is addressed. The proposed construction is mainly based on the works of Cuntz and [A. Hawkins, A. Skalski, S. White and J. Zacharias, On spectral triples on crossed products arising from equicontinuous actions, Math. Scand. 113(2) (2013) 262–291], and on our previous papers [V. Aiello, D. Guido and T. Isola, Spectral triples for noncommutative solenoidal spaces from self-coverings, J. Math. Anal. Appl. 448(2) (2017) 1378–1412; V. Aiello, D. Guido and T. Isola, A spectral triple for a solenoid based on the Sierpinski gasket, SIGMA Symmetry Integrability Geom. Methods Appl. 17(20) (2021) 21]. The embedding of [Formula: see text] in [Formula: see text] can be considered as the dual form of a covering projection between noncommutative spaces. A main assumption is the expansiveness of the endomorphism, which takes the form of the local isometricity of the covering projection, and is expressed via the compatibility of the Lip-norms on [Formula: see text] and [Formula: see text].

scholarly journals Epis and monos which must be isos

1984 ◽  
Vol 7 (3) ◽  
pp. 507-512
Author(s):  
David J. Fieldhouse
Keyword(s):  
Chain Condition ◽  
Noetherian Module ◽  
Injective Modules ◽  
Descending Chain Condition ◽  
Injective Endomorphism

Orzech [1] has shown that every surjective endomorphism of a noetherian module is an isomorphism. Here we prove analogous results for injective endomorphisms of noetherian injective modules, and the duals of these results. We prove that every injective endomorphism, with large image, of a module with the descending chain condition on large submodules is an isomorphism, which dualizes a result of Varadarajan [2]. Finally we prove the following result and its dual: ifpis any radical then every surjective endomorphism of a moduleM, with kernel contained inpM, is an isomorphism, provided that every surjective endomorphism ofpMis an isomorphism.

STABLE EQUIVALENCE PROBLEMS FOR FREE ALGEBRAS WITH THE NIELSEN-SCHREIER PROPERTY

2001 ◽  
Vol 11 (06) ◽  
pp. 779-786 ◽  
Author(s):  
ALEXANDER A. MIKHALEV ◽  
JIE-TAI YU
Keyword(s):  
Free Algebra ◽  
Maximal Rank ◽  
Variety Of Algebras ◽  
Free Algebras ◽  
Stable Equivalence ◽  
Two Systems ◽  
Equivalence Problems ◽  
Injective Endomorphism

A variety of algebras is said to be Schreier if any subalgebra of a free algebra of this variety is free in the same variety of algebras. For free algebras of finite ranks of Schreier varieties we prove that if two systems of elements are stably equivalent, then they are equivalent. We define the rank of an endomorphism of a free algebra of a Schreier variety and prove that an injective endomorphism of maximal rank does not change the rank of elements of maximal rank.

P-injective endomorphism rings

1996 ◽  
Vol 24 (12) ◽  
pp. 3759-3769 ◽  
Author(s):  
Soumaya M. Khuri
Keyword(s):  
Endomorphism Rings ◽  
Injective Endomorphism
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