The fineness properties of Morita contexts

2021 ◽  
pp. 2250205
Author(s):  
Yiqiang Zhou
Keyword(s):  
Sufficient Condition ◽  
Morita Context ◽  
Matrix Rings ◽  
Simple Ring

Let [Formula: see text] be a Morita context. For generalized fine (respectively, generalized unit-fine) rings [Formula: see text] and [Formula: see text], it is proved that [Formula: see text] is generalized fine (respectively, generalized unit-fine) if and only if, for [Formula: see text] and [Formula: see text], [Formula: see text] implies [Formula: see text] and [Formula: see text] implies [Formula: see text]. Especially, for fine (respectively, unit-fine) rings [Formula: see text] and [Formula: see text], [Formula: see text] is fine (respectively, unit-fine) if and only if, for [Formula: see text] and [Formula: see text], [Formula: see text] implies [Formula: see text] and [Formula: see text] implies [Formula: see text]. As consequences, (1) matrix rings over fine (respectively, unit-fine, generalized fine and generalized unit-fine) rings are fine (respectively, unit-fine, generalized fine and generalized unit-fine); (2) a sufficient condition for a simple ring to be fine (respectively, unit-fine) is obtained: a simple ring [Formula: see text] is fine (respectively, unit-fine) if both [Formula: see text] and [Formula: see text] are fine (respectively, unit-fine) for some [Formula: see text]; and (3) a question of Cǎlugǎreanu [1] on unit-fine matrix rings is affirmatively answered.

scholarly journals Acceptable Morita Contexts for Semigroups

ISRN Algebra ◽  
2012 ◽  
Vol 2012 ◽  
pp. 1-5 ◽  
Author(s):  
Valdis Laan
Keyword(s):  
Short Note ◽  
Morita Equivalence ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Morita Context ◽  
Necessary And Sufficient

This short note deals with Morita equivalence of (arbitrary) semigroups. We give a necessary and sufficient condition for a Morita context containing two semigroups S and T to induce an equivalence between the category of closed right S-acts and the category of closed right T-acts.

The p.p. property of trivial extensions

2015 ◽  
Vol 14 (08) ◽  
pp. 1550124 ◽  
Author(s):  
M. Tamer Koşan
Keyword(s):  
Trivial Extension ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Morita Context ◽  
Necessary And Sufficient

In this note, a necessary and sufficient condition for a trivial extension to be a left p.p.-ring is obtained. As an application, we determine when a trivial Morita context is a left p.p.-ring, and a result of Xue [On p.p. rings, Kobe J. Math. 7 (1990) 77–80] characterizing triangular p.p.-rings follows immediately.

scholarly journals Pseudo-distributive near-rings

1975 ◽  
Vol 12 (3) ◽  
pp. 449-456 ◽  
Author(s):  
Henry E. Heatherly ◽  
Steve Ligh
Keyword(s):  
Group Rings ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Matrix Rings ◽  
Basic Properties ◽  
Necessary And Sufficient

In the study of the theory of rings, matrix rings, group rings, algebras, and so on, play a very important role. However, the analogous systems may not exist in the theory of near-rings. Recently Ligh obtained a necessary and sufficient condition for the set of n × n matrices with entries from a near-ring to be a near-ring. This opens the door for the study of other structures such as group near-rings, algebras, and so on. In this paper we initiate a study of the basic properties of pseudo-distributive near-rings, which is exactly the class of near-rings needed to carry out the construction of matrix near-rings, group near-rings, polynomials with near-ring coefficients, and so on.

scholarly journals Hereditary Semi-Primary Rings and Tri-Angular Matrix Rings

1966 ◽  
Vol 27 (2) ◽  
pp. 463-484 ◽  
Author(s):  
Manabu Harada
Keyword(s):  
Matrix Ring ◽  
Global Dimension ◽  
Homological Dimension ◽  
Matrix Rings ◽  
Hereditary Ring ◽  
Simple Ring ◽  
Dimension Zero ◽  
Dimension One ◽  
Global Homological Dimension

It is well known that the semi-simple rings with minimum conditions coincide with the rings of global homological dimension zero and that the hereditary rings coincide with the rings of global dimension one. Eilenberg, Jans, Nagao and Nakayama gave some properties of hereditary rings in [4] and [11], which relate to global dimension of factor rings. As an example of non-commutative hereditary ring we know a tri-angular matrix ring over a semi-simple ring.

scholarly journals Projectivity and duality in a Morita context

1993 ◽  
Vol 47 (1) ◽  
pp. 95-100 ◽  
Author(s):  
Zhou Zhengping
Keyword(s):  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Morita Context ◽  
Closed Submodule ◽  
Left And Right ◽  
Necessary And Sufficient ◽  
Special Case

The concepts of closed submodule, left and right annihilators are generalised and a necessary and sufficient condition on any Morita context is given so that the duality and projectivity between these sets exist. As a special case, we get Hutchinson's necessary and sufficient condition under a weaker hypothesis on the context.

The Current Situation in Chemical Stabilization of Biological Materials

1972 ◽  
Vol 30 ◽  
pp. 132-133
Author(s):  
John H. Luft
Keyword(s):  
Electron Microscopy ◽  
Electron Microscope ◽  
Osmium Tetroxide ◽  
Molecular Level ◽  
Cross Linking ◽  
Chemical Stabilization ◽  
Specimen Preparation ◽  
Sufficient Condition ◽  
Radio Telescopes

With information processing devices such as radio telescopes, microscopes or hi-fi systems, the quality of the output often is limited by distortion or noise introduced at the input stage of the device. This analogy can be extended usefully to specimen preparation for the electron microscope; fixation, which initiates the processing sequence, is the single most important step and, unfortunately, is the least well understood. Although there is an abundance of fixation mixtures recommended in the light microscopy literature, osmium tetroxide and glutaraldehyde are favored for electron microscopy. These fixatives react vigorously with proteins at the molecular level. There is clear evidence for the cross-linking of proteins both by osmium tetroxide and glutaraldehyde and cross-linking may be a necessary if not sufficient condition to define fixatives as a class.

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