Monomial Ideal Theory Questions

Question

Let $R=k[x_1,\ldots,x_n]$ be a polynomial ring. Let $I,J$ be monomial ideals.

Definition: $(I:x_j)=\{f \in R \mid fx_j \in I \}$

Questions:

1. $((I+J):x_j)=(I:x_j)+(J:x_j)$ ? for some $x_j$

2. Can we write $(I^s:x_j)$ in terms of powers of $I$?

I think first one is true but i do not know second one.

Thanks

Answer 1

For the first one, let $I=(m_i)$ and $J=(n_i)$, with implied summation over $i$. Then $(I: x_j)=(m_i/\gcd(m_i, x_j))$. So $((I+J): x_j)=(m_i/\gcd(m_i, x_j), n_i/\gcd(n_i, x_j))$ and $$((I: x_j)+(J: x_j))=((m_i/\gcd(x_j)+(n_i/\gcd(n_i,x_j)))=(m_i/\gcd(m_i, x_j), n_i/\gcd(n_i, x_j))$$ so assuming the right side of your equation is the ideal generated by the set theoretic sum, we have equality.

Likewise $$(I^s:x_k)=((\prod_{i=1}^sm_{\sigma(i)}):x_k)=((\prod_{i=1}^sm_{\sigma(i)})/\gcd(m_l,x_k)=(\prod_{i=1}^{s-1}m_{\sigma(i)}))(m_{\sigma(l)}/\gcd(m_l,x_k)=I^{s-1}(I: x_k)$$