Rings and Modules

Let M be an additively written group (with neutral element 0). 

The D-group M is called unital if there exists an element d in D 0 such that, for each element m in M, 

An (additively written) group is called a ring if it is a group over itself. 

Assuming D to be a ring we have an operation of D on D or, what is the same, an operation on D in the sense of Section 5.5. This operation on D is called the multiplication of II, and one stays with the convention that be denotes the image of (b, c) under the multiplication on D whenever b and c are elements in D. 

A subgroup C of the additive group of D is called a subring if, with respect to the restriction of the multiplication on D to C x C, C is a C -group. 

The (commutative) group M is called a group over D oi D-group if, for any three elements k, I in M and d in D, 

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V. I. Amautov et ai. Introduction to the Theory of Topological Rings and Modules... 

Another group of conjugated thiophene molecules for future appHcations are those being developed as nonlinear optical (NLO) devices (75). Replacement of benzene rings with thiophene has an enormous effect on the molecular nonlinearity of such molecules. These NLO molecules are able to switch, route, and modulate light. Technology using such materials should become available by the turn of the twenty-first century. 

Systematic modification of the sulfanilamide molecule in order to maximize the hypoglycemic activity led to the observation that the sulfonamide is best replaced by a sulfonylurea function. Modification on both the aromatic ring and the substituent on the terminal nitrogen modulates the activity of the products. ... 

Rabiei, P. Steier, W. H. Zhang, C. Dalton, L. R., Polymer micro ring filters and modulators,... 

Important improvement in the in vitro activity was obtained when the central proline was replaced by thiaproline (thiaPRO) to give compound 66, suggesting that modification of the central amino acid (of the proline type) could be of importance in modulating the PEP inhibitory activity, Eq. (25) [76]. In fact, replacement of the proline moiety of63 - 66 by non-natural amino acids derived from 2-perhydroindole or from 2-azabicyclo[2.2.2]octane and modulation of the side chain by replacement of the terminal phenyl ring by a dicyclopropyl-carbinyl moiety afforded derivatives such as 67 and 68 with improved activities (IC50 between 10 and 20 nM), Eq. (26). 

Pop or modulating action available O-ring and soft seals limiting chemical and temperature compatibility... 

Since rings as well as modules are built from commutative groups and groups are identified with thin schemes (via the group correspondence) one may view the notion of a ring and the one of a module as part of thin scheme theory . 

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