Let R={1 € Q[2] : [0) € Z}. (a) Show that R is an integral domain and R* = {+1}. (b) Show that irreducibles of Rare Ep for primes pe Z, and S() ER with (0 €{+1} which are irreducible in Q[r]. (c) Show that r is not a product of irreducibles, and hence R does not satisfy the ascending chain condition for principal ideals.
5.104. Let p and q be irreducible elements of a PID R. Prove that R/(pq) = R/(p) x R/(q) if and only if p and q are not associates.
Please explain steps taken and why 21.21 Let R be a PID and I an ideal of R. a) Show that every ideal of R/I is principal. Must R/I be a PID? b) Show that R/I has only finitely many ideals if I is nontrivial.
This question is about "Real Analysis". Start with a set S c R and successively take its closure, the complement of its closure, the closure of that, and so on: S, cl(S), (cl(S))c,.... Do the same to Se. In total, how many distinct subsets of R can be produced this way? In particular decide whether each chain S, cl(S), consists of only finitely many sets. For example, if S = Q then we get Q, R, , , R, R,...
Let f:R->S be a homomorphism of rings and let K=(r in R]f(r)=0}. Prove that Khas the absorption property O If rand s are in K then f(r)=f(s)=0 so f(rs)=f(r)f(s)=0 x 0=0 O If ris in R and s is in K then f(s)=0 so f(rs)=f(r)f(s)=f(r) x 0=0 Olf rand s are in R then f(r)=f(s)=0 so f(rs)=f(r)f(s)=0 x 0=0 O If ris in R and s is in K then f(r)=f(s)=0 so f(rs)=f(r)f(s)=0 x 0=0
Let R and S be PIDs, and assume that R is a subring of S. Assume the following about R and S: If, for an element , there exists a non-zero with , then . Show: If is a greatest common divisor in S for two elements a and b in R (not both 0), then d is a greatest common divisor for a and b in R. sES TER We were unable to transcribe this imageWe were unable to...
Let R be a ring, let S be a subring of R and let' be an ideal of R. Note that I have proved that (5+1)/1 = {5 +1 | 5 € S) and I defined $:(5+1) ► S(SO ) by the formula: 0/5 + 1)=5+(SNI). In the previous video I showed that was well-defined. Now show that is a ring homomorphism. In other words, show that preserves both ring addition and ring multiplication. Then turn your work into this...
3. (20 pts) Let ụ be a finite set, and let S = {Si, S , S,n} be a collection of subsets of U. Given an integer k, we want to know if there is a sub-collection of k sets S' C S whose union covers all the elements of U. That is, S k, and Us es SU. Prove that this problem is NP-complete. 992 m SES, si 3. (20 pts) Let ụ be a finite set, and let...
Let w be a subspace of R", and let wt be the set of all vectors orthogonal to W. Show that wt is a subspace of R" using the following steps. a. Take z in wt, and let u represent any element of W. Then zu u = 0. Take any scalar c and show that cz is orthogonal to u. (Since u was an arbitrary element of W, this will show that cz is in wt.) b. Take z,...
(a) Let R be a commutative ring. Given a finite subset {ai, a2, , an} of R, con- sider the set {rial + r202 + . . . + rnan I ri, r2, . . . , rn є R), which we denote by 〈a1, a2 , . . . , Prove that 〈a1, a2, . . . , an〉 įs an ideal of R. (If an ideal 1 = 〈a1, аг, . . . , an) for some a,...