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Math and science::Algebra::Aluffi

Rings. Zero-divisors.

Zero-divisor

An element \( a \) in a ring \( (R, +, \cdot) \) is a left zero-divisor iff there exists an element \( b \neq 0 \) in \( R \) such that [\( \quad ? \;\; = \;\; ? \quad \)].

An element \( a \) in a ring \( (R, +, \cdot) \) is a right zero-divisor iff there exists an element \( b \neq 0 \) in \( R \) such that [\( \quad ? \;\; = \;\; ? \quad \)].

Lambda perspective

The left case describes the 2-input function \( \groupMul{R}: R \times R \to R \) having the first parameter fixed at \( a \). This forms a function \( m_a : R \to R \). Lambda calculus would call this partial application.

Injectivity and surjectivity

Not a left zero-divisor iff left-multiplication is injective

The partial application of \( \groupMul{R} : R \times R \to R \) by \( a \) as the first argument is an injective function iff \( a \) is not a left zero-divisor.

Can you recall the proof?

When a ring has finite elements, all of which are not zero-divisors, it is a [what?].