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

Expert Solution

Colby Messinger answered 1 year ago

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 ].

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).

If R is a division ring, show that cent R forms a field.

Prove the following: (a) Let A be a ring and B be a field. Let f...

Prove the following: (a) Let A be a ring and B be a field. Let f : A → B be a surjective homomorphism from A to B. Then ker(f) is a maximal ideal. (b) If A/J is a field, then J is a maximal ideal.

. 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.

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 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.

If F is a field and ?(?),?(?),h(?) ∈ ?[?]; and h(?) ≠ 0. a) Prove that...

If F is a field and ?(?),?(?),h(?) ∈ ?[?]; and h(?) ≠ 0. a) Prove that [?(?)] = [?(?)] if and only if ?(?) ≡ ?(?)(???( (h(?)). b) Prove that congruence classes modulo h(?) are either disjoint or identical.

Prove that {f(x) ∈ F(R, R) : f(0) = 0} is a subspace of F(R, R)....

Prove that {f(x) ∈ F(R, R) : f(0) = 0} is a subspace of F(R, R). Explain why {f(x) : f(0) = 1} is not.

A theorem for you to prove: Let B be a local ring containing a perfect field...

A theorem for you to prove: Let B be a local ring containing a perfect field k that is isomorphic to its residue field B/m, and such that B is a localization of a finitely generated k-algebra. Then the module of relative differential forms M_B/k is a free B-module of rank equal to the dimension of B if and only if B is a regular local ring.

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