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Math and science::Algebra::Aluffi

The uniqueness of the zero ring

A ring is the zero ring iff \( 0 = 1\)

That is to say, the zero ring is the only ring where both operations share the same element for their identity.

The proof is on the reverse.


Proof. The forward case is trivial, as there is only one element. So consider the reverse case. Let \( a \) be an element of the ring \( R \). Let \( e \) be the shared identity element. By \( e \) acting as a multiplicative identity we have:

\[ a = a \cdot e \]

By \( e \) acting as the additive identity we have:

\[ a \cdot (e + e) = a \cdot (e + e + e + e + e) \]

But by the distributive law, this means that:

\[ a = a + a \]

which can only be true for the additive identity. So \( a = e \) and the group must be is the zero group as \( a \) was arbitrary.