When encountered with the word 'decomposition', we immediately think of a structure being broken down into and written as a (kind of) 'union' of simpler parts(say for example the decomposition of a number into its prime factors).
But, primary decomposition is described in terms of 'intersection'. (Statement: A primary decomposition of an ideal $I$ in $R$ is the expression of $I$ as a finite intersection of primary ideals)
Is the nomenclature of this phenomenon as a 'decomposition' marely a coincidence? Or is it in line with what is traditionally meant by a decomposition?