Question 3.(a) Prove that every ring with prime characteristic p can be embedded in a ring with unityand characteristic p.(b) Let (G, +) be an Abelian group. If n and are homomorphisms from G to G, define n+Cand no by(n+C)(a) = n(a) + ((a)andnos(a) = n(5(a))for all a E G. Prove that n + C and no C are also endomorphisms. Also prove that withthese operations the set of all endomorphisms constitutes a ring.

(a) Let R be a ring with prime characteristic p. The objective is to prove that it can be embedded…

Answered: (a) Prove that every ring with prime…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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An endomorphism of the group G is a homomorphism : G → G. Let End (A) be the set of all endomorphisms of the abelian groups A, that is End(A) := {y: A → Alya homomorphism} Define binary operations + and * on End(A) by (+)(a) = y(a) + (a) and (*)(a) = ((a)) 4 for all a E A and p, & € End(A). Show that (End (A), +, *) is a ring.
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4. Write out the addition and multiplication tables for each congruence class ring below, then determine if it is a field. (a) Z3[r]/(x² +1) (b) Z₂[r]/(x² +1)
Suppose that f (x) is a fifth-degree polynomial that is irreducible overZ2. Prove that x is a generator of the cyclic group (Z2[x]/<f (x)>)*.
Let R be a nonzero ring and Endz (R) := {f : R→ R |f an additive group homomorphism}. Show that Endz (R) is a ring with addition defined by (f + g) given by (ƒ + g)(x) = f(x) + g(x) for all x E R and multiplication given by composition. Prove that the units in Endz (R) is Autz (R), the group of additive group automorphisms of R.
Concise Reason (t) If F is a field with 32 elements, and x is a non-zero and non-identity element of F, then what are the possibilities for the smallest positive integer n with x" = 1? Concise Reason u) Let R = Z[x] be the ring of polynomials in x with integer coefficients. Then does the subgroup generated by the polynomial p(x) = x + 1 in the abelian group (R, +) (which in group theory, we would denote by (x + 1)) have the same elements as the ideal generated by p(x) = x + 1 in the commutative ring (R,+,) (which in ring theory, we would denote by (x + 1))? Concise Reason Have a good summer!
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(19) Let om : Z → Zm be the ring homomorphism defined by om (a) remainder of division of a by m. (a) Show that m : Z[x] → Zm[x] defined by om (anx + + a₁x + ao) = om(an)x² + is a ring homomorphism onto Zm[x]. + om (a₁)x+om (ao) = (b) Show that if f(x) = Z[x] and ¯m (f(x)) = Zm[x] both have degree n and σm(f(x)) does not factor in Zm[x] into two polynomials of degree less than n, then f(x) is irreducible Q[x]. (c) Use the previous part to show that x³ + 17x + 36 is irreducible in Q[x].

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