Understanding the definition of t IJ when I AND J are ideals

Question

if I is ideal and J is ideal then

$$IJ=\left\{ \displaystyle\sum\limits_{i=1}^n a_i b_i\mid a_i\in I, b_i\in J \right\}$$

i am not undetsnaging the defination of it, what is n some n that belong to N? n = |I| or |J|? according to this definitiation it look like that |I| = |J| so each ideals inside some ring have the same size?

Yes, $n\in\mathbb N$. No, $n$ has nothing to do with the size of $I$ or $J$, it ranges over all integers. I don't know why you think it means $|I|=|J|$. Both of them could be infinite. You should also add "$n\in\mathbb N$" inside the braces. — rschwieb, Jun 30 '17 at 13:56
It words, this says "The set of all finite sums of products $ij$ where $i\in I$ and $j\in J$" — rschwieb, Jun 30 '17 at 13:58
Form a set consisting of all the possible products of elements of $I$ with elements of $J$: $$P={ab;|;a\in I, b\in J}.$$ Form another set that consists of all the possible finite sums of elements of the set $P$. What you then get is the set $IJ:$ $$IJ=\left{\sum_{finite}ab ;|; ab\in P\right}$$ — Prajwal Kansakar, Jun 30 '17 at 16:13

score 0 · Accepted Answer · answered Jun 30 '17 at 13:56

0

Maybe this looks clearer: $$ \bigcup_{n=1}^\infty\left\{a_1b_1+\cdots+a_nb_n: a_1,\ldots,a_n \in I \,\text{ and }\,b_1,\ldots,b_n \in J\right\} $$

answered Jun 30 '17 at 13:56

Paolo Leonetti

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again from your defination it look like |I| = |j|, what about define it in case when I,J are finite? and has diffrent sizes? – ido kahana Jun 30 '17 at 13:59
No. The definition has nothing to do with the cardinalities of $I$ and $J$, finite or not, equal or not. – Paolo Leonetti Jun 30 '17 at 14:00
@idokahana Do you think the $a_i$ are somehow an exhaustive list of elements of $I$? It isn't. – rschwieb Jun 30 '17 at 14:03

Understanding the definition of t IJ when I AND J are ideals

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