Algebraic element

An algebraic element is invertible iff its minimal polynomial has a nonzero constant term

Let be a [[K-monoid|-ring]] and be an algebraic element with minimal polynomial . Then the following are equivalent¹

1. is invertible in
2. is not a left Zero-divisor
3. is not a right Zero-divisor
4. is not a (two-sided) Zero-divisor
5. has a nonzero constant term, i.e. .

Proof

If is invertible but for some , then and thus , a contradiction. For the converse, if and , then

so

is the inverse.

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Footnotes

1. 2008. Advanced Linear Algebra, §18, pp. 459–461 ↩