Exercise 23, § 7

a) Let $A$ be a local noetherian ring such that e(A) = 1. Prove that $A$ possesses a single minimal prime ideal p such that dim(A) = dim(A/p). Prove that moreover one has long(A_p) = 1 and e(A/p) = 1 (§ 7, No. 1, remark 4). b) Recall (p. 82, Exercise 10) that a local ring is equidimensional if, for every minimal prime ideal q of $A$, one has dim(A/q) = dim(A) and that it has no embedded prime ideals if the $A$-module A_s does not possess embedded associated prime ideals (IV, § 2, No. 3, remark). (It will be seen in chapter X that the completions of integral quotients of local Macaulay rings possess these two properties.) \ Let $A$ be a local, noetherian, equidimensional ring without embedded prime ideals, such that e(A) = 1. Prove that $A$ is integral.*