Let $R=k[x\_1,\ldots,x\_n]$ be a graded ring and $S,T,U$ be monomials ideals.
$reg(S)=max\{j-i \backslash \beta\_{i,j}(S) \neq 0\}$.


Assume $S+T=U$


**prove \disprove :** $reg(S+T^2) \leq reg(U^2)$.


We can see $reg(S+T^2) \leq reg(S)+reg(T^2)-1 < reg(S)+reg(T^2)$ by [By Herzog result, see Corollary 3.2](http://arxiv.org/pdf/math/0512021v1.pdf)



Castelnuovo- Mumford regularity properties
This example is due to Aldo Conca:


$R = k[x\_1,x\_2,x\_3], S = (x\_1^3x\_2,x\_1x\_2^3,x\_2^4,x\_1^2x\_2^2x\_3^5), T = (x\_1^4), U = S+T$. Then $\text{reg} (S+T^2) = 9$, while $\text{reg} U^2 = 8$.


