Thus, for example, the studies of " hypercomplex numbers ", such as considered by the Quaternion Society, were put onto an axiomatic footing as branches of ring theory ( in this case, with the specific meaning of associative algebras over the field of complex numbers . ) In this context, the quotient ring concept is one of the most powerful unifiers.
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The definition of an ideal is such that the ideal " I " generated ( see below ) by " Z " is exactly the set of elements that are forced to become zero if " Z " becomes zero, and the quotient ring " R / I " is the desired ring where " Z " is zero, and " only " elements that are forced by " Z " to be zero are zero.
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