Componentwise Linear Ideals From Sums

Let $I,J$ be componentwise linear ideals in a polynomial ring $S$. We study necessary and sufficient conditions for $I+J$ to be componentwise linear. We provide a complete characterization when $\dim S=2$. As a consequence, any componentwise linear monomial ideal in $k[x,y]$ has linear quotients using generators in non-decreasing degrees. In any dimension, we show that under mild compatibility conditions, one can build a componentwise linear ideal from a given collection of componentwise linear monomial ideals using only sum and product with square-free monomials. We provide numerous examples to demonstrate the optimality of our results.