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Unit sum numbers of right self-injective rings

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Chapters 3, 4 and 5

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... An ordered pair or couple (a, b) is an object having two entries, coordinates or projections, where the first or left entry, is distinguishable from the second or right entry. For example, (a, b) is distinguishable from (b, a) unless a = b. Perhaps the best example of an ordered pair is (x, y) that ...
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MATH 361: NUMBER THEORY — TENTH LECTURE The subject of

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... (iii) SR; Q is certainly a ring using operations of R, so it’s a subring, but it isn’t an ideal. √ the usual√ / Q. (Any ideal containing 1 must be the whole For instance, 1 ∈ Q and 2 ∈ R, but 2 ∈ ring!) (iv) SR; The sum, difference, or product of two 2π-periodic functions is still 2π-periodic, and 0 ...
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Artinian and Noetherian Rings

... Theorem is in the realm of modules. It is this application that we will discuss in this section. The Artinian and Noetherian framework that we discussed above for rings can be generalized to encompass modules, another type of algebraic structure. In simple terms, modules are like vector spaces but o ...
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... 1. The kernel always contains the identity element e, so its non-empty. Let a, b be elements in the kernel. We have that ab−1 is in the kernel since Φ(ab−1 ) = Φ(a)Φ(b)−1 = e · e−1 = e. Therefore the kernel is a subgroup. We need to check normality. Let x be any element of G. The element xax−1 is ma ...
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... (That is, the preimage of an open set is open.) We will require later the following result. Theorem S 1.3 (Heine-Borel) If [a, b] ⊆ R is covered by a collection of (ci , di ), so [a, b] ⊆ i∈I (ci , di ), then there exists a finite sub-collection of the (ci , di ), S which can relabeled as 1 ≤ i ≤ N ...
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Ring Theory

< 1 ... 19 20 21 22 23 24 25 26 27 ... 39 >

Commutative ring

In ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra.Some specific kinds of commutative rings are given with the following chain of class inclusions: Commutative rings ⊃ integral domains ⊃ integrally closed domains ⊃ unique factorization domains ⊃ principal ideal domains ⊃ Euclidean domains ⊃ fields ⊃ finite fields
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