在交换代数中，准素分解将一个交换环的理想（或模的子模）唯一地表成准素理想（或准素子模）之交。这是算术基本定理的推广，能用以处理代数几何中的情况。

设 为交换诺特环， 为有限生成之 -模。对任一子模 ，存在有限多个准素子模 使得

事实上，可以要求此分解是最小的（即：无法省去任何 ），且诸准素子模 对应到的素理想彼此相异。满足上述条件的准素分解是唯一确定的。

最常见的情形是取 ，并取 为一理想。任取一准素分解 ，这些 中的极小者称为 的孤立素理想，否则称为镶嵌素理想；孤立素理想是 的一组不变量。

In mathematics, the Lasker–Noether theorem states that every Noetherian ring is a Lasker ring, which means that every ideal can be decomposed as an intersection, called primary decomposition, of finitely many primary ideals (which are related to, but not quite the same as, powers of prime ideals). The theorem was first proven by EmanuelLasker (1905) for the special case of polynomial rings and convergent power series rings, and was proven in its full generality by EmmyNoether (1921).