Algebraic extensions of semirings

Dear all,

I am interested in algebraic extensions of commutative semirings.
Can somebody tell me why the literature almost exclusively seems to
deal with algebraic extensions of fields? What's so special about
them? We can write down polynomials over loads of other structures.

In particular, I'm interested in the semiring of pairs of positive
real numbers (a,b) with a<=b, with multiplication and addition
performed elementwise. What would an algebraic extension of this
produce? Presumably it would give a ring, as we require the
polynomials x+(a,b)=0 to have a solution. Unfortunately, I really do
want a ring, not a semiring --- is there any natural way to weaken the
notion of algebraic extension so that semirings aren't turned into
rings? Or perhaps, I have completely the wrong end of the stick here,
in which case, I'd be glad for somebody to correct me.

Regards,
Jamie.

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