Question

Let R be a commutative ring, with ideals I and J. Let I + J = {a + b: a ? I, b ? J}. Prove that I + J is an ideal of R. For obvious reasons, we call the ideal I + J the sum of the ideals I and J. Suppose that R is a commutative ring with unity, and I and J are ideals. Define I · J = {?_{k=1}^n a_k b_k : a_k ? I, b_k ? J, n ? ?}.

          Let R be a commutative ring, with ideals I and J. Let
I + J = {a + b: a ? I, b ? J}.
Prove that I + J is an ideal of R. For obvious reasons, we call the ideal I + J the sum of the ideals I and J.
Suppose that R is a commutative ring with unity, and I and J are ideals. Define
I · J = {?_{k=1}^n a_k b_k : a_k ? I, b_k ? J, n ? ?}.

Let R be a commutative ring, with ideals I and J. Let
I + J = a + b: a ? I, b ? J.
Prove that I + J is an ideal of R. For obvious reasons, we call the ideal I + J the sum of the ideals I and J.
Suppose that R is a commutative ring with unity, and I and J are ideals. Define
I · J = ?k=1^n ak bk : ak ? I, bk ? J, n ? ?.

Added by George B.

Calculus: Early Transcendentals

James Stewart 8th Edition

Instant Answer

Solved by Expert Sri K

02/27/2023

Step 1

We need to show that $I+J$ is an ideal, which means it is closed under addition and multiplication by elements in $R$. Show more…

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Let R be a commutative ring with ideals, and let I + J = {a + b: a ∈ I, b ∈ J}. Prove that I + J is an ideal of R. For obvious reasons, we will call the ideal I + J the sum of the ideals. Suppose that R is a commutative ring with unity, and let I and J be ideals. Define I · J = {∑_{k=1}^n a_k b_k : a_k ∈ I, b_k ∈ J, n ∈ N}.