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2 order is too weak an invariant

Every (associative, unital) ring is a subring of the endomorphism ring of its underlying additive group. Rings act on abelian groups; groups act on sets. The universal action on an abelian group is its endomorphism ring; the universal action on a set is the symmetric group. Modules are rings that remember their action on an abelian group; permutation groups are groups that remember their action on a set.

A set is determined by its cardinality, but for abelian groups cardinality is not a very useful invariant. Rather than "order" of a ring, consider the isomorphism class of its underlying additive group. This is even commonly done in the finite ring case, where the order still has some mild control, but not as much as the isomorphism type of the additive group.

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Every (associative, unital) ring is a subring of the endomorphism ring of its underlying additive group. Rings act on abelian groups; groups act on sets. The universal action on an abelian group is its endomorphism ring; the universal action on a set is the symmetric group. Modules are rings that remember their action on an abelian group; permutation groups are groups that remember their action on a set.