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I need literature about zeroes of polynomials and equation resolution in associative algebraic structures with zero-divisors, but I am having difficulties to find it.

For example, there is literature in the resolution of quadratic(and remarks on other degrees) in quaternions and split-quaternions.

It seems that search engines do something with the keyword zero, that has two meanings in this context, this is, zero-divisor and zeroes of the polynomials.

So far I do not find ways to more concrete examples. It would be fine also, algebraic literature ( as opposed to specific examples )

Thanks in advance

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For associative algebras, as your required, see Plotkin, Algebras with the same (algebraic) geometry, Israel J. Math., 96 (2) (1996), 511–522.

This is, being more precise, part of this nice relatively new field of Universal Algebraic Geometry which discuss such things.

For a survey I recommend A. Shevlyakov, Lectures notes in universal algebraic geometry, https://arxiv.org/abs/1601.02743.

EDIT depending on the kind of question you are interested in, pherhaps the theory of (associative) PI-algebras might be also interesting. For this I recommend V. Drensky's Free Algebras and PI-algebras.

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  • $\begingroup$ I forget to add other details, specifically, the property of multiplicative inverse for nonzero elements. In anycase, I will examine the literature that you had provided ;) $\endgroup$
    – Dragon lala lalo
    Commented Jul 22, 2020 at 18:46

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