Multiplicative monoid modulo n

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Definition

Let n be a positive integer. The multiplicative monoid modulo n is defined as follows:

  • Its underlying set is the set \{ 0,1,2,\dots,n-1\}.
  • The product of two elements a,b in the set, is defined by the rule: multiply them as integers, and then take the remainder of the product modulo n.

Alternatively, the multiplicative monoid modulo n can be defined as the monoid of congruence classes mod n under multiplication.

Facts

  • The multiplicative monoid modulo n is a monoid of size n.
  • The multiplicative monoid modulo n has identity element (neutral element) 1 and zero element (nil element) 0. It is not a group.
  • The multiplicative monoid modulo n is Abelian: any two elements in it commute.