Question

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.

Answer #1

Let R be an integral domain. Prove that R[x] is an integral
domain.

Let R be an integral domain. Prove that if R is a field to begin
with, then the field of quotients Q is isomorphic to R

6.4.7.
(a) Show that a subring R' of an integral domain R is an
integral
domain, if 1âˆˆ R'.
(b) The Gaussian integers are the complex numbers whose real
and
imaginary parts are integers. Show that the set of Gaussian
integers
is an integral domain.
(c) Show that the ring of symmetric polynomials in n variables is
an
integral domain.

Prove that every field is an integral domain. Give an example of
an integral domain that is not a field. Give an example of a ring
that is not an integral domain.

Consider the ring R = Q[x]/<x^2>. (a) Is R an integral
domain? Justify your answer. (b) IS [x+1] a unit in R? If it is,
find its multiplicative inverse.

Suppose R is a Principle Ideal Domain and a and b are nonzero
elements of R. Prove that (a,b)=(d) if and only if d is a greatest
common divisor of a and b.

Prove or disprove (a) Z[x]/(x^2 + 1), (b) Z[x]/(x^2 - 1) is an
Integral domain.
By showing (a) x^2+1 is a prime ideal or showing x^2 + 1 is not
prime ideal.
By showing (b) x^2-1 is a prime ideal or showing x^2 - 1 is not
prime ideal.
(Hint: R/I is an integral domain if and only if I is a prime
ideal.)

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)

Prove that there is no integral domain with exactly 6 elements.
Can your argument be adapted to show that there is no integral
domain with exactly 4 elements? What about 15 elements? Use these
observations to guess a general result about the number of elements
in a finite integral domain.

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