Number Sets and Algebra
... 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 ...
... 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 ...
Solutions Sheet 7
... 5. Let X be a locally noetherian scheme. Prove that the set of irreducible components of X is locally finite, i.e. that every point of X has an open neighborhood which meets only finitely many irreducible components of X. Solution: By definition the irreducible components of a topological space are ...
... 5. Let X be a locally noetherian scheme. Prove that the set of irreducible components of X is locally finite, i.e. that every point of X has an open neighborhood which meets only finitely many irreducible components of X. Solution: By definition the irreducible components of a topological space are ...
![(January 14, 2009) [08.1] Let R be a principal ideal domain. Let I be](http://s1.studyres.com/store/data/006005520_1-5192ab5794c8b33a00720a831d1a8a33-300x300.png)
Solution
... (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 ...
... (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 ...
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 ...
... 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 ...
Solutions
... 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 ...
... 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 ...
Notes 1
... (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 ...
... (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 ...
