injectivity condition
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2018 ◽  
Vol 59 (2) ◽  
pp. 021902 ◽  
Author(s):  
Andras Molnar ◽  
Yimin Ge ◽  
Norbert Schuch ◽  
J. Ignacio Cirac

2016 ◽  
Vol 15 (08) ◽  
pp. 1650152 ◽  
Author(s):  
Yasser Ibrahim ◽  
Xuan Hau Nguyen ◽  
Mohamed F. Yousif ◽  
Yiqiang Zhou

It is well known that if every cyclic right module over a ring is injective, then the ring is semisimple artinian. This classical theorem of Osofsky promoted a considerable interest in the rings whose cyclics satisfy a certain generalized injectivity condition, such as being quasi-injective, continuous, quasi-continuous, or [Formula: see text]. Here we carry out a study of the rings whose cyclic modules are [Formula: see text]-modules. The motivation is the observation that a ring [Formula: see text] is semisimple artinian if and only if every [Formula: see text] -generated right [Formula: see text]-module is a [Formula: see text]-module. Many basic properties are obtained for the rings whose cyclics are [Formula: see text]-modules, and some structure theorems are proved. For instance, it is proved that a semiperfect ring has all cyclics [Formula: see text]-modules if and only if it is a direct product of a semisimple artinian ring and finitely many local rings, and that a right self-injective regular ring has all cyclics [Formula: see text]-modules if and only if it is a direct product of a semisimple artinian ring, a strongly regular ring and a [Formula: see text] matrix ring over a strongly regular ring. Applications to the rings whose [Formula: see text]-generated modules are [Formula: see text] -modules, and the rings whose cyclics are ADS or quasi-continuous are addressed.


2005 ◽  
Vol 15 (02) ◽  
pp. 217-254 ◽  
Author(s):  
J. HYNDMAN ◽  
J. G. PITKETHLY

We show that, within the class of three-element unary algebras, there is a tight connection between a finitely based quasi-equational theory, finite rank, enough algebraic operations (from natural duality theory) and a special injectivity condition.


2004 ◽  
Vol 14 (6) ◽  
pp. 657-668 ◽  
Author(s):  
SERGIO ANTOY ◽  
MICHAEL HANUS

An injective finite mapping is an abstraction common to many programs. We describe the design of an injective finite mapping and its implementation in Curry, a functional logic language. Curry supports the concurrent asynchronous execution of distinct portions of a program. This condition prevents passing from one portion to another a structure containing a partially constructed mapping to ensure that a new choice does not violate the injectivity condition. We present some motivating problems and we show fragments of programs that solve these problems using our design and implementation.


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