Given $I,J$ ideals of $R$, show that $\forall m \geq 1; I^m + J^m =R$

Question

Given $R$ a commutative ring and $I,J$ ideals of $R$, such that $I+J=R$, show that $I^m+J^m=R, \forall m \geq 1$

My problem is that I don't know what is the meaning of $I^m$ and in literature I didn't find anything. I know that, given $A,B$ ideals, then $AB= \{ \sum_{i=1}^n a_ib_i |a_i \in A, b_i \in B, n \in \mathbb{N} \}$, but with this notation $I^2=II=I$, so what is the meaning of $I^m$? It could be $I^m = \{ x^m | x \in I \} $?

You got your $AB$ wrong. Note that if $r \in R$ and $a \in A$, then $r a \in A$. So what you have written is the ideal $A + B$. Instead, $A B$ is the ideal spanned by the products $a b$, for $a \in A$ and $b \in B$. Similarly, $I^{m}$ is the ideal spanned by all $m$-fold products $x_{1} \cdots x_{m}$, for $x_{i} \in I$. — Andreas Caranti, Jan 26 '16 at 21:20
Nope, your definition is still wrong. $AB=\sum a_ib_i$, not product. — Thomas Andrews, Jan 26 '16 at 21:24
Thank you for the corrections, and for the clarification about $I^m$, any hint for the exercise? — HaroldF, Jan 26 '16 at 21:24

score 6 · Accepted Answer · answered Jan 26 '16 at 21:26

6

The definition of $I^m$ is recursive: $I^1=I$ and $I^{m+1}=II^m$, where the right side is the product of ideals.

Hint: Note that $1=i+j$ for some $i\in I,j\in J$. Then expand $1=(i+j)^{2m-1}$ to see that $1\in I^m + J^m$.

answered Jan 26 '16 at 21:26

Thomas Andrews

177,126

if I expand $1= (i+j)^{2m-1}= i^{2m-1}+j^{2m-1}+$"terms in $i^kj^{2m-1-k}"$ Now I have that the terms in $i^kj^{2m-1-k}$ lies in $IJ$ (and also their sum) but then? – HaroldF Jan 26 '16 at 21:37

Given $I,J$ ideals of $R$, show that $\forall m \geq 1; I^m + J^m =R$

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