Joint work with Amartya Goswami. Let A be a commutative ring and let I be an ideal. An element a∈A is a regular denominator for I when multiplication by a on A/I is injective; we denote the set of all such elements by SI. We study how the common regular denominators for two ideals I and J are related to I∩J and I+J. This yields a particularly simple description when I and J are comaximal. Over Noetherian rings, the same viewpoint classifies denominator-equivalence classes by finite nonempty antichains of prime ideals, gives bounds for the associated primes of an intersection and leads to a local length identity. Furthermore, we show that flat base change preserves regular denominators, faithful flatness reflects them, and finite locally free quotients have fiberwise regular loci defined by determinants satisfying a multiplicative Mayer-Vietoris formula. Our results provide alternatives to primary-decomposition computations in many concrete situations. Download [here](https://arxiv.org/abs/2608.21102) 