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

Please show all parts of answer

Solutions

Expert Solution

Colby Messinger answered 1 year ago

Next > < Previous

Related Solutions

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...

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 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.

Let L = {0 r | r = s 2 , s a positive integer}. Give...

Let L = {0 r | r = s 2 , s a positive integer}. Give the simplest proof you can that L is not regular using the pumping lemma.

Let T: R^2 -> R^2 be an orthogonal transformation and let A is an element of...

Let T: R^2 -> R^2 be an orthogonal transformation and let A is an element of R^(2x2) be the standard matrix of T. In a) and b) below, by rotation we mean "rotation of R^2 by some angle and the origin". By reflection, we mean "reflection of R^2 over some line through the origin". a) Show that T is either a rotation or a reflection. b) Show that every rotation is a composition of 2 reflections, and thus that T...

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 x be the number of fouls more than (i.e., over and above) the opposing team....

Let x be the number of fouls more than (i.e., over and above) the opposing team. Let y be the percentage of times the team with the larger number of fouls wins the game. x 0 4 5 6 y 49 43 33 26 Complete parts (a) through (e), given Σx = 15, Σy = 151, Σx2 = 77, Σy2 = 6015, Σxy = 493, and r ≈ −0.906. (a) Draw a scatter diagram displaying the data. (b) Verify the...

L = {a r b s | r, s ≥ 0 and s = r 2}....

L = {a r b s | r, s ≥ 0 and s = r 2}. Show that L is not regular using the pumping lemma

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)) >...

Subjects