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Is it always the case that the additive identity annihilates all elements under multiplication? I can't think of an example; my course essentially relies on integers, polynomials, and matrices for most examples of any given concept. I'm curious if there is an example where this is not the case (and avoid erroneous proofs in exercises!)

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$0 \cdot x = (0+0)\cdot x = 0 \cdot x + 0 \cdot x \implies 0 \cdot x = 0$

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On one hand $(a + 0)\cdot b = a \cdot b$

but on the other hand, by the distributive property $(a + 0)\cdot b = a\cdot b + 0 \cdot b$

so equating the two, we necessarily have $0\cdot b = 0$.

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Yes, that is true.

For any $a\in R$, $0 \cdot a = (1 - 1)\cdot a = 1\cdot a - 1\cdot a = 0$.

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