Question

Assume R is a commutative ring. (a) Prove that the intersection of two ideals in a ring R is also an ideal. (b) If I, J are ideals of a ring R, define I + J = {i + j : i ? I, j ? J}. Prove that I + J is the smallest ideal containing I ? J. Find 8? + 12?. (c) Let I, J be ideals of R. Define the product IJ = {?_{k=1}^{n} i_k j_k : i_k ? I, j_k ? J, n ? ?_+}. Prove that IJ is an ideal contained in the ideal I ? J. Find IJ and I ? J when I = 8? and J = 12?.

          Assume R is a commutative ring.
(a) Prove that the intersection of two ideals in a ring R is also an ideal.
(b) If I, J are ideals of a ring R, define I + J = {i + j : i ? I, j ? J}. Prove that I + J is the smallest ideal containing I ? J. Find 8? + 12?.
(c) Let I, J be ideals of R. Define the product
IJ = {?_{k=1}^{n} i_k j_k : i_k ? I, j_k ? J, n ? ?_+}.
Prove that IJ is an ideal contained in the ideal I ? J. Find IJ and I ? J when I = 8? and J = 12?.

Assume R is a commutative ring.
(a) Prove that the intersection of two ideals in a ring R is also an ideal.
(b) If I, J are ideals of a ring R, define I + J = i + j : i ? I, j ? J. Prove that I + J is the smallest ideal containing I ? J. Find 8? + 12?.
(c) Let I, J be ideals of R. Define the product
IJ = ?k=1^n ik jk : ik ? I, jk ? J, n ? ?+.
Prove that IJ is an ideal contained in the ideal I ? J. Find IJ and I ? J when I = 8? and J = 12?.

Added by Antonio A.

Calculus: Early Transcendentals

James Stewart 8th Edition

Instant Answer

Solved by Expert Sri K

11/05/2022

Step 1

Step 1: To prove that the intersection of two ideals \( I \) and \( J \) in a commutative ring \( R \) is also an ideal, we need to show two properties: closure under addition and closure under multiplication by any element of \( R \). Show more…

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Assume R is a commutative ring. (a) Prove that the intersection of two ideals in a ring R is also an ideal. (b) If I, J are ideals of a ring R, define I + J = {i + j : i ∈ I, j ∈ J}. Prove that I + J is the smallest ideal containing I ∪ J. Find 8ℤ + 12ℤ. (c) Let I, J be ideals of R. Define the product IJ = {∑(k=1 to n) ikjk : ik ∈ I, jk ∈ J, n ∈ ℤ+}. Prove that IJ is an ideal contained in the ideal I ∩ J. Find IJ and I ∩ J when I = 8ℤ and J = 12ℤ.