Question

Let $I$ and $J$ be ideals in $R$. Let $IJ$ denote the set of all possible finite sums of elements of the form $ab$ (with $a in I, b in J$), that is, $IJ = {a_1b_1 + a_2b_2 + dots + a_nb_n mid n ge 1, a_k in I, b_k in J}.$ Prove that $IJ$ is an ideal, $IJ$ is called the product of $I$ and $J$.

Name: Let I and J be ideals in R. Let IJ denote the set of all possible finite sums of elements of the form ab (with a in I, b in J), that is, IJ = a1b1 + a2b2 + dots + anbn mid n ge 1, ak in I, bk in J. Prove that IJ is an ideal, IJ is called the product of I and J.
Uploaded: 2024-02-11T22:53:55-08:00
Duration: 2 min 34 s
Channel: Mauya Mitchell
Description: Let I and J be ideals in R. Let IJ denote the set of all possible finite sums of elements of the form ab (with a in I, b in J), that is, IJ = a1b1 + a2b2 + dots + anbn mid n ge 1, ak in I, bk in J. Prove that IJ is an ideal, IJ is called the product of I and J.

          Let $I$ and $J$ be ideals in $R$. Let $IJ$ denote the set of all possible finite sums of elements of the form $ab$ (with $a in I, b in J$), that is, $IJ = {a_1b_1 + a_2b_2 + dots + a_nb_n mid n ge 1, a_k in I, b_k in J}.$ Prove that $IJ$ is an ideal, $IJ$ is called the product of $I$ and $J$.

Let I and J be ideals in R. Let IJ denote the set of all possible finite sums of elements of the form ab (with a in I, b in J), that is, IJ = a1b1 + a2b2 + dots + anbn mid n ge 1, ak in I, bk in J. Prove that IJ is an ideal, IJ is called the product of I and J.

Added by Jessica S.

Calculus: Early Transcendentals

James Stewart 8th Edition

Instant Answer

Solved by Expert Mauya Mitchell

02/11/2024

Step 1

To show that IJ is closed under addition, consider two elements x = a1b1 + a2b2 + ... + an-1bn-1 and y = c1d1 + c2d2 + ... + cm-1dm-1 in IJ. Then, x + y = (a1b1 + a2b2 + ... + an-1bn-1) + (c1d1 + c2d2 + ... + cm-1dm-1) can be expressed as a finite sum of elements Show more…

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Let I and J be ideals in R. Let IJ denote the set of all possible finite sums of elements of the form ab (with a in I, b in J), that is, IJ = {a1b1 + a2b2 + ... + anbn | n >= 1, ak in I, bk in J}. Prove that IJ is an ideal. IJ is called the product of I and J.