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 [ $? = ?$ ].

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 [ $? = ?$ ].

Lambda perspective

The left case describes the 2-input function $\cdot_{R} : R \times R \to R$ 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 $\cdot_{R} : R \times R \to R$ 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?].

29.06.2022