Structural Theory of Milnor Squares

There are numerous papers focused on pullbacks, among which Milnor squares are particularly significant in applications. Their success in generating intriguing examples has garnered substantial attention in multiplicative ideal theory. A notable example is Seidenberg’s construction, which delineates all possible relationships between the Krull dimension of a ring R and that of its polynomial ring R[X]. This chapter systematically compiles results related to Milnor squares, with most of the material drawn from the literature [86, 98, 102, 214].