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To allow for Division By Zero, we develop a new algebraic structurecontaining addition and multiplication called an S-Extension of a Field. Thisunique structure extends a Field so that the equation $0\cdot s=x$ has exactlyone solution for every non-zero Field element $x$. Furthermore, a differentsolution is obtained for each choice of $x$, making this solution unique tothat particular equation. However, the equation $0\cdot s=0$ has two or moresolutions, with no preference towards any one particular solution. This allowsus 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 ${x\over0}$ is a unique element that is also unique to that particular $x$ while${0\over 0}$ remains indeterminate. This creates a Division By Zero whichsignificantly differs from other attempts at Division By Zero.

A New Algebraic Structure That Extends Fields And Allows For A True Division By Zero