# Math Help - Additive order of elements in Euclidean Rings

1. ## Additive order of elements in Euclidean Rings

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2. Originally Posted by abstracto
I'm stumped by this problem. Any help would be greatly appreciated

Show that the additive order of a Euclidean ring is either prime, infinite, or one, where the additive order of g in (R,+,*) is the order of g in group (R,+)

Also, if anyone can help me find all the group homomorphisms from the cyclic group Z12 to alternating group A5, that would be awesome too.

you don't need R to be Euclidean. the claim is true in any domain because in every domain all non-zero elements have the same additive order, which is the charateristic of the domain.

for the second question just note that a group homomorphism $f$ is completely determined by $f(1)=\sigma,$ where $\sigma$ is any element of $A_5$ whose order is a divisor of 12. so we must have

$o(\sigma) \in \{1,2,3,4,6,12 \}.$ but $A_5$ has no element of order 4, 6 or 12 and it has 20 elements of order 3 and 15 elements of order 2. so there are 36 group homomorphisms from $\mathbb{Z}_{12}$ to $A_5.$