3. Suppose R is a PID and M is a cyclic R-module of 'order' r E R, ie., M RI (r). Show that if N ...
13.12.8 Problem. Let R be a ring and, let M be an R-module. Let m be a nonnegative integer, and suppose that M1,..., Mm are R-submodules of M, and that M is the internal direct sum of M1,..., Mm. Let n be a nonnegative integer with n < m, and for each i E {1,...,n}, let N; be an R-submodule of M. Let N = N1 ++ Nn. ... (i) Prove that N is the internal direct sum of N1,...,...
Let M be a 8:27 AM right R-module, N be an (R,T)-bimodule, and L be a left T-module. Let e: (MN)* L M R (NB, L) be given by e (moon, e) = m (nol). Let m.con, mone MORN, and lEl. Prove e (lm, BR.) + (m₂ Ore), d)= e(m, on, d) + (mon, e). This is the proof I'm working on. I need to show the map I've defined (and which is defined towards the middle of the proof)...
Theorem 4.27. Suppose G is a finite cyclic group of order n. Then G is isomorphic to Rn if n ≥ 3, S2 if n = 2, and the trivial group if n = 1. Most of the previous results have involved finite cyclic groups. What about infinite cyclic groups?
4. Suppose G is a group of order n < 0. Show that if G contains a group element of order n, then G is cyclic.
(Abstract Algebra) Please answer a-d clearly. Show your work and explain your answer. (a) Let G be a group of order 4 with identity e. Show that G is either cyclic or a2-e for all (b) Does the result of part (a) generalize to groups of order p2 for any positive integer p? In other words, is it the case that if G is a group of order p2 with identity e, then is either cyclic or a- e for...
multi and optimization, please help Problem 3: Let ri be given n mutually orthogonal vectors in R", and zo E R" be also given. Find (a) the distance di from zo to H, := {TE Rn : ХТХǐ (b) the distance sk from zo to tiHi, 1-k < n (c) the distance mk from xo to k+iHi,1 S k < n (d) calculate sk + mk. 0) Problem 3: Let ri be given n mutually orthogonal vectors in R", and...
Suppose that d = s and and positive integers m and n (a) Show that m/d and n/d are relatively prime ged(m, n) sm +tn for some integers (b) Show that if d = s'm + t'n for s', t' e Z, then s' = s kn/d for some k e Z.
Third Order (a-i, n = 3 ): It has been said that when a process is more than second order, that a PID might be workable, but insufficient. Please explain why this might be the case. Delay-Dominated First Order: Control of systems with large time delays can be difficult consider avariation on P with largedeľay ( TD,TD> a) ( (s) = T s It is has been said that for such systems that adding derivative action does not help much....
3.) Let ak E R with ak > 0 for all k E N. Suppose Σ㎞iak converges. Show that Σί1bk (By definition, for a sequence (ck), we say liCkoo if, for all M ER with Hint: Show that there exists (Ni))ไ1 with N > Nj for all j E N, such that bk there exists a sequence (bk)k of real numbers such that lim converges = oo and M >0, there exists N E N such that ck > M...
(a) Suppose A is an n x n real matrix. Show that A can be written as a sum of two invertible matrices. HINT: for any le R, we can write A = XI + (A - XI) (b) Suppose V is a proper subspace of Mn,n(R). That is to say, V is a subspace, and V + Mn.n(R) (there is some Me Mn,n(R) such that M&V). Show that there exists an invertible matrix M e Mn,n(R) such that M&V....