By R.S. Pierce

For lots of humans there's lifestyles after forty; for a few mathematicians there's algebra after Galois concept. the target ofthis ebook is to end up the latter thesis. it really is written essentially for college students who've assimilated significant parts of a regular first 12 months graduate algebra textbook, and who've loved the event. the cloth that's awarded the following shouldn't be deadly whether it is swallowed via people who're now not contributors of that workforce. The items of our consciousness during this ebook are associative algebras, commonly those which are finite dimensional over a box. This topic is perfect for a textbook that may lead graduate scholars right into a really expert box of study. the key theorems on associative algebras inc1ude probably the most perfect result of the good heros of algebra: Wedderbum, Artin, Noether, Hasse, Brauer, Albert, Jacobson, and so on. the method of refine ment and c1arification has introduced the evidence of the gemstones during this topic to a degree that may be preferred through scholars with simply modest history. the topic is nearly certain within the wide variety of contacts that it makes with different components of arithmetic. The research of associative algebras con tributes to and attracts from such subject matters as staff concept, commutative ring thought, box idea, algebraic quantity thought, algebraic geometry, homo logical algebra, and type thought. It even has a few ties with components of utilized arithmetic.

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**Example text**

I(X) = X, 2. i(S) ⊂ S, 3. i(i(S)) = i(S), 4. i(S ∩ T ) = i(S) ∩ i(T ). 12. Let (X, τ ) be a topological space. Then the mappings intτ , clτ : P(X) → P(X) are interior and closure operators, respectively. Proof. Obviously, intτ (X) = X and intτ (A) ⊂ A. Moreover, intτ (U ) = U for U ∈ τ , and intτ (A) ∈ τ , because τ is a topology. Hence intτ (intτ (A)) = intτ (A). Finally, intτ (A ∩ B) ⊂ intτ (A) ⊂ A, intτ (A ∩ B) ⊂ intτ (B) ⊂ B, yielding intτ (intτ (A ∩ B)) ⊂ intτ (intτ (A) ∩ intτ (B)) ⊂ intτ (A ∩ B), where intτ (A) ∩ intτ (B) ∈ τ , so that intτ (A ∩ B) = intτ (A) ∩ intτ (B).

1 (Continuous mappings). A mapping f : X → Y is continuous at point x ∈ X if x ∈ c(A) =⇒ f (x) ∈ c (f (A)) for every A ⊂ X. , f (c(A)) ⊂ c (f (A)) for every A ⊂ X. If precision is needed, we may emphasize the topologies involved and, instead of mere continuity, speak speciﬁcally about (τX , τY )-continuity. The set of continuous functions from X to Y is often denoted by C(X, Y ), with convention C(X) = C(X, R) (or C(X) = C(X, C)). 10. 2. Let c ∈ R. Let f, g : X → R be continuous, where we use the Euclidean metric topology on R.

Let (X, d) be a metric space and A ⊂ X. Prove the following claims: intd (A) = {x ∈ X | ∃r > 0 : Br (x) ⊂ A} , ∂d (A) = cld (A) \ intd (A), X = intd (A) ∪ ∂d (A) ∪ extd (A). Consequently, prove that cld (A) is closed for any set A ⊂ X. 36 Chapter A. 3 (Metric topology). Let (X, d) be a metric space. Then τd := intd (P(X)) = {intd (A) | A ⊂ X} is called the metric topology or the family of metrically open sets. The corresponding family of metrically closed sets is τd∗ := cld (P(X)) = {cld (A) | A ⊂ X} .