Exercise 8, § 9

Let A be a ring, M an A-module, $x' = (x_1, ..., x_p)$ and $x'' = (x_{p+1}, ..., x_n)$ two sequences of elements of A; denote by x the sequence $(x_1, ..., x_n)$, and by M’ the A-module $M/(x')$ M. (a) If x’ is completely secant for M and x” completely secant for M’, now prove that x is completely secant for M. (b) If the sequences x and x’ are completely secant for M, now prove that the sequence x” is completely secant for M’. (c) Give an example of an A-regular sequence (x, y) contained in the radical of A, such that x is completely secant for A/yA, but that y is not completely secant for A.