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Abstract algebra, the un-codified implications of which need to be fully understood at school to gain good grades in exams.  School algebra and numeracy is all about persuading pupils that the numbers we have work by these rules, and not by the various coping patterns they learned earlier in their education.

Why do we need Z as an axiom; can’t we conclude it from the others? If a is non-zero, then it has an inverse by M4, and then (1/a) ∙ a ∙ b = (1/a) ∙ 0, so b = 0; thus either a or b must be 0. (We need the fact that x ∙ 0 = 0 for any x, but this follows by observing that x ∙ 0 = x ∙ (0+0) = x ∙ 0 + x ∙ 0, and adding -(x ∙ 0) to both sides.)

Yes you are quite correct.  I suppose that it is in my list above to enable the axioms to be switched on and off in different combinations to allow for, say, a ring with no zero divisors; but I know that that is not what I am taking about here.  And yet I like it there from a school maths point of view.  Pupils should use this property when solving quadratic equations by factorization and, along with say -1x-1=+1, this may be one of the opportunities for a more profound algebraic discussion in the class room.  In my experience they may understand the algebra, or just do what they are told without being bothered why, or stumble over the fact not finding it obvious.  It is this third set who might be deep thinkers and could be very rewarding to teach maths to.

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