Let I1, I2 be ideals of R and J1, J2 be ideals of S. Show that...

Let I1, I2 be ideals of R and J1, J2 be ideals of S. Show that (I1 + I2)^extension= I1^extension + I2^extension where I1, I2 are contained in R

|^e is defined as the extension of I to S: Let R and S be commutatuve ring and f:R to S be a ring homomorphism. For each ideal I of R, the ideal f(I)S of S generated by f(I) is the extension of I to S.

Expert Solution

I've shown both side contentments.

Colby Messinger answered 1 month ago

10.3.6 Exercise: Product of Pairwise Comaximal Ideals. Let R be a commutative ring, and let {A1,...,An}...

10.3.6 Exercise: Product of Pairwise Comaximal Ideals. Let R be a commutative ring, and let {A1,...,An} be a pairwise comaximal set ofn ideals. Prove that A1 ···An = A1 ∩ ··· ∩ An. (Hint: recall that A1 ···An ⊆ A1 ∩···∩An from 8.3.8).

Let A be such that its only right ideals are {¯0} (neutral element) and A. Show...

Let A be such that its only right ideals are {¯0} (neutral element) and A. Show that A or is a ring with division or A is a ring with a prime number of elements in which a · b = 0 for any, b ∈ A.

5. Show that if R is a division ring,then Mn(R) has no nontrivial two-sided ideals.

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 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 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 R be a ring and n ∈ N. Let S = Mn(R) be the ring...

Let R be a ring and n ∈ N. Let S = Mn(R) be the ring of n × n matrices with entries in R. a) i) Let T be the subset of S consisting of the n × n diagonal matrices with entries in R (so that T consists of the matrices in S whose entries off the leading diagonal are zero). Show that T is a subring of S. We denote the ring T by Dn(R). ii). Show...

. 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.

Let L = {x = a r b s c t | r + s =...

Let L = {x = a r b s c t | r + s = t, r, s, t ≥ 0}. Give the simplest proof you can that L is not regular using the pumping lemma.

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