  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.
