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Suppose I have a field-like structure with a set $F$ and two operations (addition and multiplication) and their respective inverses. It respects the following proprieties similar to a field:

  • Associativity of $+$ and $\times$;
  • Commutativity of $+$ and $\times$;
  • Distributivity of $\times$ over $+$;
  • A multiplicative identity $I$;
  • Existence of additive and multiplicative inverses;

Notice how there is no mention of an additive identity. Furthermore, there exists a set $S$ containing "base" elements of $F$ (all elements of $F$ are derived from additions and multiplications of elements in $S$). For an element $A$ of $S$, $A+A^2$ doesn't make sense. You can only add elements that have the same power. Let's suppose there exists an additive identity $k$. This would entail the following:

$$ A+k = A \;\therefore\; A+(A^2-A^2) = A \;\therefore\; (A+A^2)-A^2 = A $$

Since we're adding elements with different powers, this equation suddenly doesn't make sense. To fix this, we can add the following axioms:

$$ a\neq b \implies k^a\neq k^b,\forall a,b\in\mathbb Z $$ $$ A^n - A^n = k^n$$

Does an algebraic structure like this present any inconsistencies?

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    $\begingroup$ It doesn't make sense to talk about inverses if you have no identity. If anything, you can talk about "division". $(a-b)$ could still make sense, but not just $-b$. $\endgroup$ – tomasz Apr 19 '14 at 15:11
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    $\begingroup$ You lost me: if there's no additive "identity", how can there be additive inverses, or better said: what's their meaninng?? $\endgroup$ – DonAntonio Apr 19 '14 at 15:11
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    $\begingroup$ You cannot have additive inverses without an additive identity. Also you can't have binary operations $+$ and $\times$ if $A+A^2$ doesn't make sense. $\endgroup$ – Seth Apr 19 '14 at 15:12
  • $\begingroup$ Also, you might be interested in the concept of graded algebra. I think it would be helpful if you stated some motivation for this concept. $\endgroup$ – tomasz Apr 19 '14 at 15:14
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    $\begingroup$ Without an additive identity, additive inverses are undefined. Without additive inverses, subtraction is undefined. By definition, the symbol $A-B$ means $A + (-B)$, where $-B$ is the additive inverse of $B$. $\endgroup$ – MPW Apr 19 '14 at 15:21

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