Question

Prove that every nonzero element of Zn is either a unit or a zero divisor, but not both.

Answer #1

(§2.3) (a) If a is a unit in Zn, prove that a is not a zero
divisor in Zn.
(b) If a is a zero divisor in Zn, prove that a is not a unit in
Zn.

Prove that if x ∈ Zn − {0} and x has no common divisor with n
greater than 1, then x has a multiplicative inverse in (Zn − {0},
·n).
State the theorem about Euler’s φ function and show why this
fact implies it.

An element [a] of Zn is said to be idempotent if [a]^2 = [a].
Prove that if p is a prime number, then [0] and [1] are the only
idempotents in Zp. (abstract algebra)

An element [a] of Zn is said to be idempotent if [a]^2 = [a].
Prove that if p is a prime number, then [0] and [1] are the only
idempotents in Zp. (abstract algebra)

Given that R is an integral domain, prove that
a) the only nilpotent element is the zero element of R,
b) the multiplicative identity is the only nonzero idempotent
element.

Prove that if A is a nonsingular nxn matrix, then so is cA for
every nonzero real number c.

Prove that every nonzero coefficient of the Taylor series of (1
- x + x²) eˣ

Prove every element of An is a product of
3-cycles.

26.9 Prove that if eventsAand B are disjoint and both have
nonzero probability, then they are dependent

(2) Letn∈Z+ withn>1. Provethatif[a]n
isaunitinZn,thenforeach[b]n ∈Zn,theequation[a]n⊙x=[b]n has a unique
solution x ∈ Zn.
Note: You must find a solution to the equation and show that
this solution is unique.
(3) Let n ∈ Z+ with n > 1, and let [a]n, [b]n ∈ Zn with
[a]n ̸= [0]n. Prove that, if the equation [a]n ⊙ x = [b]n has no
solution x ∈ Zn, then [a]n must be a zero divisor.

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