# integer?

#### alexandrabel90

what does it mean by integer/ m integer?

im asked to find all the units of integer/ 12 integer and im not sure what does that mean.

thanks

Rapha

#### HallsofIvy

MHF Helper
what does it mean by integer/ m integer?

im asked to find all the units of integer/ 12 integer and im not sure what does that mean.

thanks
I'm not certain what you mean by "integer/12 integer" but I suspect you mean $$\displaystyle \mathbb{Z}_{12}$$, the "integers modulo 12". The "units of a ring" are those members of ring that have a multiplicative inverse. It should be easy to see that if a number has a common factor with 12, then some multiple of it will be a multiple of 12, so congruent to 0 modulo 12, so cannot have a multiplicative inverse.

#### alexandrabel90

does it mean that for example in integer modulo 12, [2] will not have an inverse since gcd of 12 and 2 is not 1?
so units of a ring are [5], [7],[11]?

#### HallsofIvy

MHF Helper
More to the point, [2] does not have an inverse (and so is not a unit) because [2][6]= [0]. Of course, that is a direct result of the fact that gcd(2, 12)= 2 not 1.

[5], [7], and [11] are units but you are missing one obvious one. Remember that the set of all units of a ring form a group with multiplication.

#### alexandrabel90

hmmm i dont get what does it mean by the set of all units of a ring form a group with multiplication. does it mean that [5],[7],[11] are cyclic groups that will generate interger 12?