integer?

Aug 2009
639
2
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
 
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HallsofIvy

MHF Helper
Apr 2005
20,249
7,909
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.
 
Aug 2009
639
2
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
Apr 2005
20,249
7,909
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.
 
Aug 2009
639
2
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?