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walter r July 10th, 2014 11:58 AM

Demonstration involving ideals of a ring
 
Please, check my solution to the following exercise:

"Let $I \subset A$ and $J \subset A$ be ideals of $A$. Show that:
a) $I \cap J$ is an ideal of A.
b) $I+J=\left\{x+y;x \in I, y \in J \right \}$ is an ideal of $A$.


a) since $I$ and $J$ are ideals, then $I \neq \varnothing$ and $J \neq \varnothing$. It implies that $I \cap J \neq \varnothing$.
$a,b \in I \cap J \rightarrow a,b \in I$ and $a,b \in J \rightarrow a+b \in I \cap J$ and $ab \in I \cap J$. Hence $I \cap J$ is an ideal.

b) $I \neq \varnothing$ and $J \neq \varnothing \rightarrow I+J \neq \varnothing$. $a,b \in I+J \rightarrow a= x+y; b= v+w; x,v \in I; y,w \in J$. But then $a+b =(x+v)+(y+w) \in I + J$, because $(x+v) \in I$ and $(y+w) \in J$.
Additionaly, if $a,b \in I+J$ then $ab= xv+xw+yv+yw$ with $xv \in I; yw \in J; xw \in I \cap J; yv \in I \cap J$. Then $ab=(xv+xw)+(yv+yw) \in I+J$. Hence $I+J$ is an ideal.
Thanks!

Deveno July 10th, 2014 12:55 PM

The intersection of two non-null sets is not necessarily non-null. For example, if $A = \{1\}$ and $B = \{2\}$, then $A \cap B = \emptyset$.

However, there is always ONE element of $A$ guaranteed to be in $I$ AND $J$. Can you think of what it is?

Again, just because two elements of two sets are in both of them, does not guarantee that the sum of these two elements is in EITHER set. Suppose we take:

$A = \{1,2,4\}$ and $B = \{1,2,5\}$.

Then $1,2 \in A \cap B$, but $1 + 2 = 3$ is not in EITHER set.

You are failing to invoke the proper properties of ideals in your proof. You're just stating that what you HOPE to be true IS true. That just won't do.


THIS is what you have to prove for (a):

1. $I \cap J$ is an additive subgroup of $A$.
2. If $a \in A$, and $x \in I \cap J$, that $ax,xa \in I \cap J$.

I'll show you how a proof of (2) starts.

Let $a \in A$, and $x \in I \cap J$.

Since $I \cap J \subseteq I$, we have $x \in I$. Since $I$ is an ideal, $ax \in I$.

What do you suppose comes next?

walter r July 11th, 2014 06:28 AM

Good morning, Deveno! Thanks for writing. Concerning item a:
Quote:

The intersection of two non-null sets is not necessarily non-null. For example, if A={1} and B={2}, then A∩B=∅.

However, there is always ONE element of A guaranteed to be in I AND J. Can you think of what it is?
(1) Do you mean $0 \in I, 0\in J \rightarrow 0 \in I \cap J$?

Quote:

Again, just because two elements of two sets are in both of them, does not guarantee that the sum of these two elements is in EITHER set.
At this point I disagree.
(2) $a,b \in I \cap J \rightarrow a,b \in I$ and $a,b \in J$. But $I,J$ are ideals, therefore $a+b \in I$ and $a+b \in J$. Hence $a+b \in I \cap J$. This is not true for any set, but it is true for an ideal.
After reading your post I redo the other step the following way:
(3) $a \in I \cap J, n \in A \rightarrow a \in I, a \in J, n \in A \rightarrow na \in I, na \in J$ (by hypothesis that both I and J are ideals)$ \rightarrow na \in I \cap J$.

(1),(2),(3) together imply $I \cap J$ is an ideal, similarly for item b. Do you think now it is convincent?

Deveno July 11th, 2014 07:19 AM

Yes, that is what was missing: you need to invoke the CLOSURE properties of ideals. This may have been what you were thinking, but until you write it down, no one else knows that.


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