Question

Suppose S is a ring with p elements, where p is prime.

a)Show that as an additive group (ignoring multiplication), S is cyclic.

b)Show that S is a commutative group.

Answer #1

1(a) Suppose G is a group with p + 1 elements of order p , where
p is prime. Prove that G is not cyclic.
(b) Suppose G is a group with order p, where p is prime. Prove
that the order of every non-identity element in G is p.

Suppose that R is a commutative ring without
zero-divisors.
Let x and y be nonzero elements.
1. Suppose that x has infinite additive order.
Show that y also has infinite additive order.
2. Suppose that the additive order of x is n.
Show that the additive order of y is at most n.
3.Show that all the nonzero elements of R have the same additive
order.

Show that the nilpotent elements of a commutative ring form a
subring.

Let R be a commutative ring with unity. Let A consist of all
elements in A[x] whose constant term is equal to 0. Show that A is
a prime ideal of A[x]

Suppose that R is a commutative ring without unity and also
without zero-divisors.
Show that the characteristic of R is zero or prime.

Let G be a group and let p be a prime number such that
pg = 0 for every element g ∈ G.
a. If
G is commutative under multiplication, show that the mapping
f : G → G
f(x) =
xp
is a homomorphism
b. If G is
an Abelian group under addition, show that the mapping
f : G → G
f(x) = xpis a homomorphism.

Prove that if p is prime then Zp, where it is nonzero, is a
group under multiplication.

Prove that if p is prime then Zp, where it is nonzero, is a
group under multiplication.

Let P be a commutative PID (principal ideal domain) with
identity. Suppose that there is a surjective ring homomorphism f :
P -> R for some (commutative) ring R. Show that every ideal of R
is principal. Use this to list all the prime and maximal ideals of
Z12.

12.29 Let p be a prime. Show that a cyclic group of order p has
exactly p−1 automorphisms

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