What to read after A Singular Introduction to Commutative Algebra?

From the reviews of the first edition:

"It is certainly no exaggeration to say that ... A Singular Introduction to Commutative Algebra aims to lead a further stage in the computational revolution in commutative algebra ... . Among the great strengths and most distinctive features ... is a new, completely unified treatment of the global and local theories. ... making it one of the most flexible and most efficient systems of its type....another strength of Greuel and Pfister's book is its breadth of coverage of theoretical topics in the portions of commutative algebra closest to algebraic geometry, with algorithmic treatments of almost every topic....Greuel and Pfister have written a distinctive and highly useful book that should be in the library of every commutative algebraist and algebraic geometer, expert and novice alike."

J.B. Little, MAA, March 2004

The second edition is substantially enlarged by a chapter on Groebner bases in non-commtative rings, a chapter on characteristic and triangular sets with applications to primary decomposition and polynomial solving and an appendix on polynomial factorization including factorization over algebraic field extensions and absolute factorization, in the uni- and multivariate case.