Note on gilmer’s multiplicative ideal theory, I

الملخص EN

As a branch of the commutative ring theory, we have the multiplicative ideal theory.

Also, Gilmer’s Multiplicative Ideal Theory [1] is a basic reference in multiplicative ideal theory.

We know that various terms in the theory are defined analogously for semigroups (especially, for commutative semigroups); those are ideal, integral element, common divisor, common multiple, (Krull) dimension, valuation.....We confer A.H.

Clifford and G.B.

Preston [2], which is a basic reference in the theory of semigroups.

The aim of this note is to prove or disprove all theorems in [ 1, Chapters I-V] for commutative semigroups (explicitly, for grading monoids).

We allow this note to be self-contained.

We leave two questions.