A field is a commutative ring with unity in which every nonzero element is a unit....

A field is a commutative ring with unity in which every nonzero element is a unit.

Question: Show that Z_5 under addition and multiplication mod 5 is a field. (state the operations, identities, inverses)

Expert Solution

Colby Messinger answered 1 year ago

Let R be a commutative ring with unity. Prove that f(x) is R[x] is a unit...

Let R be a commutative ring with unity. Prove that f(x) is R[x] is a unit in R[x] iff f(x)=a is of degree 0 and is a unit in R.

Let R be a commutative ring with unity. If I is a prime ideal of R,...

Let R be a commutative ring with unity. If I is a prime ideal of R, prove that I[x] is a prime ideal of R[x].

Let A be a commutative ring and F a field. Show that A is an algebra...

Let A be a commutative ring and F a field. Show that A is an algebra over F if and only if A contains (an isomorphic copy of) F as a subring.

Given a commutative ring with identity R. A. Prove that a unit is not a zero...

Given a commutative ring with identity R. A. Prove that a unit is not a zero divisor. B. Prove that a zero divisor cannot be a unit.

Let R be a commutative ring with identity with the property that every ideal in R...

Let R be a commutative ring with identity with the property that every ideal in R is principal. Prove that every homomorphic image of R has the same property.

An element a in a field F is called a primitive nth root of unity if...

An element a in a field F is called a primitive nth root of unity if n is the smallest positive integer such that an=1. For example, i is a primitive 4th root of unity in C, whereas -1 is not a primitive 4th root of unity (even though (-1)4=1). (a) Find all primitive 4th roots of unity in F5 (b) Find all primitive 3rd roots of unity in F7 (c) Find all primitive 6th roots of unity in F7...

Let R and S be commutative rings with unity. (a) Let I be an ideal of...

Let R and S be commutative rings with unity. (a) Let I be an ideal of R and let J be an ideal of S. Prove that I × J = {(a, b) | a ∈ I, b ∈ J} is an ideal of R × S. (b) (Harder!) Let L be any ideal of R × S. Prove that there exists an ideal I of R and an ideal J of S such that L = I × J.

Problem 1. Suppose that R is a commutative ring with addition “+” and multiplication “·”, and...

Problem 1. Suppose that R is a commutative ring with addition “+” and multiplication “·”, and that I a subset of R is an ideal in R. In other words, suppose that I is a subring of R such that (x is in I and y is in R) implies x · y is in I. Define the relation “~” on R by y ~ x if and only if y − x is in I, and assume for the...

say R1, R2,...., Rn are commutative rings with unity. Show that U(R1 + R2 +.... +...

say R1, R2,...., Rn are commutative rings with unity. Show that U(R1 + R2 +.... + Rn) = U( R1) + U(R2)+ .... U(Rn). Where U - is the units of the ring.

Suppose a and b belong to a commutative ring and ab is a zero-divisor. Show that...

Suppose a and b belong to a commutative ring and ab is a zero-divisor. Show that either a or b is a zero-divisor. provide explanation please.

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