For an arbitrary ring R, prove that a) If I is an ideal of R, then...

For an arbitrary ring R, prove that a) If I is an ideal of R, then I[ x] forms an ideal of the polynomial ring R[ x]. b) If R and R' are isomorphic rings, then R[ x] is isomorphic to R' [ x ].

Expert Solution

Colby Messinger answered 2 years ago

If I is an ideal of the ring R, show how to make the quotient ring...

If I is an ideal of the ring R, show how to make the quotient ring R/I into a left R-module, and also show how to make R/I into a right R-module.

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].

1. If I is an ideal of the ring R, show how to make the quotient...

1. If I is an ideal of the ring R, show how to make the quotient ring R/I into a left R-module, and also show how to make R/I into a right R-module.

prove that a ring R is a field if and only if (R-{0}, .) is an...

prove that a ring R is a field if and only if (R-{0}, .) is an abelian group

Let S ⊆ R and let G be an arbitrary isometry of R . Prove that...

Let S ⊆ R and let G be an arbitrary isometry of R . Prove that the symmetry group of G(S) is isomorphic to the symmetry group of S. Hint: If F is a symmetry of S, what is the corresponding symmetry of G(S)?

Let R be a ring with at least two elements. Prove that M2×2(R)is always a ring...

Let R be a ring with at least two elements. Prove that M2×2(R)is always a ring (with addition and multiplication of matrices deﬁned as usual).

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.

. Let φ : R → S be a ring homomorphism of R onto S. Prove...

. Let φ : R → S be a ring homomorphism of R onto S. Prove the following: J ⊂ S is an ideal of S if and only if φ ^−1 (J) is an ideal of R.

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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.

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