Let R and S be rings. Denote the operations in R as +R and ·R and...

Let R and S be rings. Denote the operations in R as +_R and ·_R and the operations in S as +S and ·S

(i) Prove that the cartesian product R × S is a ring, under componentwise addition and multiplication.

(ii) Prove that R × S is a ring with identity if and only if R and S are both rings with identity.

(iii) Prove that R × S is a commutative ring if and only if R and S are both commutative rings.

Solutions

Expert Solution

Colby Messinger answered 1 month ago

Next > < Previous

Related Solutions

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.

Let Rx denote the group of nonzero real numbers under multiplication and let R+ denote the...

Let Rx denote the group of nonzero real numbers under multiplication and let R+ denote the group of positive real numbers under multiplication. Let H be the subgroup {1, −1} of Rx. Prove that Rx ≈ R+ ⊕ H.

Let R and S be nontrivial rings (i.e., containing more than just the 0 element), and...

Let R and S be nontrivial rings (i.e., containing more than just the 0 element), and define the projection homomorphism pi_1: R X S --> R by pi_1(x,y)=x. (a) Prove that pi_1 is a surjective homomorphism of rings. (b) Prove that pi_1 is not injective (c) Prove that (R X S)/R (not congruent) R. Please show all parts of answer

2.2.6. Let S be a subset of a group G, and let S^-1 denote {s^-1: s...

2.2.6. Let S be a subset of a group G, and let S^-1 denote {s^-1: s ∈ S}. Show that 〈S^-1〉 = 〈S 〉. In particular, for a ∈ G, 〈a〉 = 〈a^-1〉, so also o(a) =o(a^-1)

The Chinese Remainder Theorem for Rings. Let R be a ring and I and J be...

The Chinese Remainder Theorem for Rings. Let R be a ring and I and J be ideals in R such that I + J = R. (a) Show that for any r and s in R, the system of equations x ≡ r (mod I) x ≡ s (mod J) has a solution. (b) In addition, prove that any two solutions of the system are congruent modulo I ∩J. (c) Let I and J be ideals in a ring R...

Let R be a ring (not necessarily commutative), and let X denote the set of two-sided...

Let R be a ring (not necessarily commutative), and let X denote the set of two-sided ideals of R. (i) Show that X is a poset with respect to to set-theoretic inclusion, ⊂. (ii) Show that with respect to the operations I ∩ J and I + J (candidates for meet and join; remember that I+J consists of the set of sums, {i + j} where i ∈ I and j ∈ J) respectively, X is a lattice. (iii) Give...

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)?

a) Let S ⊂ R, assuming that f : S → R is a continuous function,...

a) Let S ⊂ R, assuming that f : S → R is a continuous function, if the image set {f(x); x ∈ S} is unbounded prove that S is unbounded. b) Let f : [0, 100] → R be a continuous function such that f(0) = f(2), f(98) = f(100) and the function g(x) := f(x+ 1)−f(x) is equal to zero in at most two points of the interval [0, 100]. Prove that (f(50) − f(49))(f(25) − f(24)) >...

Let L = {x = a^rb^s | r + s = 1mod2}, I.e, r + s...

Let L = {x = a^rb^s | r + s = 1mod2}, I.e, r + s is an odd number. Is L regular or not? Give a proof that your answer is correct.

Due October 25. Let R denote the set of complex numbers of the form a +...

Due October 25. Let R denote the set of complex numbers of the form a + b √ 3i, with a, b ∈ Z. Define N : R → Z≥0, by N(a + b √ 3i) = a 2 + 3b 2 . Prove: (i) R is closed under addition and multiplication. Conclude R is a ring and also an integral domain. (ii) Prove N(xy) = N(x)N(y), for all x, y ∈ R. (ii) Prove that 1, −1 are the...

Subjects