A New Algebraic Structure That Extends Fields And Allows For A True   Division By Zero
  To allow for Division By Zero, we develop a new algebraic structure containing addition and multiplication called an S-Extension of a Field. This unique structure extends a Field so that the equation $0\cdot s=x$ has exactly one solution for every non-zero Field element $x$. Furthermore, a different solution is obtained for each choice of $x$, making this solution unique to that particular equation. However, the equation $0\cdot s=0$ has two or more solutions, with no preference towards any one particular solution. This allows us to use the usual definition of division as the solution to the equation $0\cdot s=x$ to evaluate $x$ divided by $0$. And if $x\not=0$, every ${x\over 0}$ is a unique element that is also unique to that particular $x$ while ${0\over 0}$ remains indeterminate. This creates a Division By Zero which significantly differs from other attempts at Division By Zero. 
Brendan Santangelo
math.GM