Communitative ring, map R / ( I /\ J) -> R/I x R/J

[Fernando Revilla]

Commutative ring, map R / ( I /\ J) -> ( R/I ) x ( R/J )

I quote an unsolved question posted in MHF (November 25th, 2012) by user needhelp2.

Say that R is a commutative ring and the I and J are ideals. Show that
the map : R=(I intersection J) maps to R/I R/J given by (r + (I intersection J)) maps to (r + I; r + J) is
well defined and is an injection. Show further more that is a surjection if and
only if I + J = R.

P.S. Communicative note: Of course I meant in the title, commutative instead of communitative.

 

[Fernando Revilla]

I suppose you mean $$\phi:R\;/\;(I\cap J) \to (R\;/\;I)\times (R\;/\;J),\;\phi(r+I\cap J)=(r+I,r+J)$$

(a) $$\phi$$ is well defined. Suppose $$r+I\cap J=r'+I\cap J$$, this implies $$r-r'\in I\cap J$$, that is $$r-r'\in I$$ and $$r-r'\in J$$. As a consequence $$r+I=r'+I$$ and $$r+J=r'+J$$ or equivalently, $$(r+I,r+J)=(r'+I,r'+J)$$: the image does not depend on the representants.

(b) $$\phi$$ is injective. Suppose $$\phi(r_1+I\cap J)=\phi(r_2+I\cap J)$$ then, $$(r_1+I,r_1+J)=(r_2+I,r_2+J)$$, hence $$r_1-r_2\in I$$, $$r_1-r_2\in J$$ or equivalently $$r_1-r_2\in I\cap J$$ which implies $$r_1+I\cap J=r_2+I\cap J$$: $$\phi$$ is injective.

(c) $$\phi$$ is a surjection $$\Leftrightarrow\; R=I+J$$.

$$\Rightarrow)$$ Let $$s\in R$$, as $$\phi$$ is a surjection there exists $$r\in R$$ such that $$\phi(r+I\cap J)=(0+I,s+J)$$, that is $$r+I=0+I$$ and $$r+J=s+J$$. This implies $$r\in I$$ and $$s-r\in J$$, so $$s=r+j$$ with $$r\in I$$ and $$j\in J$$. As a consequence $$I+J\subset R\subset I+J$$, or equivalently $$R=I+J$$.

$$\Leftarrow)$$ (Left as an exercise for the reader). :)

 

[Deveno]

suppose $R = I+J$. then for any $r \in R$ we have $r = x+y$. for some $x \in I, y \in J$.

let $(r + I,r'+J)$ be any element of $R/I \times R/J$.

writing $r = x + y, r' = x' + y'$ we have:

$r+I = (x+y)+I = (y+x)+I = y+I + x+I = y+ I + I = y + I$ and:

$r'+J = (x'+y')+J = x'+J + y'+J = x'+J + J = x' + J$

let $s = x'+y$. then

$\phi(s+(I\cap J)) = \phi((x'+y)+(I\cap J)) = ((x'+y)+I,(x'+y)+J)$

$= ((x'+I)+(y+I),(x'+J)+(y+J)) = (I+(y+I),(x'+J)+J)= (y+I,x'+J) = (r+I,r'+I)$

so $\phi$ is surjective.

oh snap! this is the chinese remainder theorem in disguise, isn't it?

 

[Fernando Revilla]

oh snap! this is the chinese remainder theorem in disguise, isn't it?

A very interesting question. :)

 

[Deveno]

If $R = \Bbb Z$ then the condition $I + J = R$ is equivalent to:

$(a) + (b) = (1)$ that is, a and b are co-prime: gcd(a,b) = 1 (using tacitly the fact that $\Bbb Z$ is a principal ideal domain, which follows from the fact that it is euclidean).

In this case, $I\cap J = (a) \cap (b) = (\text{lcm}(a,b)) = \left(\frac{ab}{\gcd(a,b)}\right) = (ab)$

We can thus conclude that if gcd(a,b) = 1:

$\Bbb Z/(ab) \cong \Bbb Z/(a) \times \Bbb Z/(b)$ a more familiar form of the CRT.