Algebra I Final Project

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Algebra I Final Project Name: Instruction: In the following questions, you should work out the solutions independently and write them down in a clear and concise manner. Staple this page as the cover sheet of your project. 1. (30 pts) (a) A pointed set is a pair (S; x) with S a set and x 2 S. A morphism of pointed sets (S; x) ! (S0; x0) is a triple (f; x; x0), where f : S ! S0 is a function such that f(x) = x0. Show that pointed sets form a category. (b) Show that in the category S∗ of pointed sets products always exist; describe them. (c) Show that in S∗ every family of objects has a coproduct (often called a \wedge product"); describe this coproduct. 2. (20 pts) Let F be the free group on a set X and G the free group on a set Y . Let F 0 be the subgroup of F generated by faba−1b−1 j a; b 2 F g and similarly for G0. Prove the following claims: 0 0 (a) F C F , and F=F is a free abelian group of rank jXj. (b) F ' G if and only if jXj = jY j. In particular, if F is also a free group on a set Z, then jXj = jZj. 3. (20 pts) Every subgroup and every quotient group of a nilpotent group is nilpotent. 4. (10 pts) Let H be a subgroup of a ﬁnite group G. Suppose that P is a Sylow p-subgroup of H. If NG(P ) ⊆ H, show that P is a Sylow p-subgroup of G. 5. (40 pts) Let R be a principal ideal domain. (a) Every proper ideal is a product P1P2 ··· Pn of maximal ideals, which are uniquely determined up to order. (b) An ideal P in R is said to be primary if ab 2 P and a 62 P imply bn 2 P for some n. Show that P is primary if and only if for some n, P = (pn), where p 2 R is prime (= irreducible) or p = 0. ni (c) If P1;P2; ··· ;Pn are primary ideals such that Pi = (pi ) and the pi are distinct primes, then P1P2 ··· Pn = P1 \ P2 \···\ Pn. (d) Every proper ideal in R can be expressed (uniquely up to order) as the intersection of a ﬁnite number of primary ideals. 6. (10 pts) If f : A ! A is an R-module homomorphism such that ff = f, then A = Ker f ⊕ Imf: 7. (20 pts) Let A1 / A2 / A3 / A4 / A5 α1 α2 α3 α4 α5 B1 / B2 / B3 / B4 / B5 be a commutative diagram of R-modules and R-module homomorphisms, with exact rows. Prove that: (a) α1 an epimorphism and α2, α4 monomorphisms ) α3 is a monomorphism; (b) α5 a monomorphism and α2, α4 epimorphisms ) α3 is an epimorphism. 1.

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Modules and Vector Spaces
Modules and Vector Spaces R. C. Daileda October 16, 2017 1 Modules Definition 1. A (left) R-module is a triple (R; M; ·) consisting of a ring R, an (additive) abelian group M and a binary operation · : R × M ! M (simply written as r · m = rm) that for all r; s 2 R and m; n 2 M satisfies • r(m + n) = rm + rn ; • (r + s)m = rm + sm ; • r(sm) = (rs)m. If R has unity we also require that 1m = m for all m 2 M. If R = F , a field, we call M a vector space (over F ). N Remark 1. One can show that as a consequence of this definition, the zeros of R and M both \act like zero" relative to the binary operation between R and M, i.e. 0Rm = 0M and r0M = 0M for all r 2 R and m 2 M. H Example 1. Let R be a ring. • R is an R-module using multiplication in R as the binary operation. • Every (additive) abelian group G is a Z-module via n · g = ng for n 2 Z and g 2 G. In fact, this is the only way to make G into a Z-module. Since we must have 1 · g = g for all g 2 G, one can show that n · g = ng for all n 2 Z. Thus there is only one possible Z-module structure on any abelian group. • Rn = R ⊕ R ⊕ · · · ⊕ R is an R-module via | {z } n times r(a1; a2; : : : ; an) = (ra1; ra2; : : : ; ran): • Mn(R) is an R-module via r(aij) = (raij): • R[x] is an R-module via X i X i r aix = raix : i i • Every ideal in R is an R-module.
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