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There’s something just a bit odd though about the notion of idempotent monoid in $Ab$. When one says that a ring is a monoid in $Ab$, one means a monoid in the monoidal category $(Ab, \otimes)$ with the standard tensor product. However, this tensor product isn’t cartesian, and so expressing the notion of an idempotent monoid (where the axiom $x x = x$ involves duplication of a variable) doesn’t go down so smoothly.
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