The quotient ring of a Boolean ring modulo a ring ideal corresponds to the factor algebra of the corresponding Boolean algebra modulo the corresponding order ideal.
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An easy motivational example is the quotient ring \ mathbb { Z } / n \ mathbb { Z } for any integer n > 1.
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For other examples of quotient objects, see quotient ring, quotient space ( linear algebra ), quotient space ( topology ), and quotient set.
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By extension, in abstract algebra, the term quotient space may be used for quotient modules, quotient rings, quotient groups, or any quotient algebra.
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The naive solution is to replace the fraction field by the total quotient ring, that is, to invert every element that is not a zero divisor.
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Hence we may localize the ring R at the set S to obtain the total quotient ring S ^ {-1 } R = Q ( R ).
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If R is a domain, then S = R-\ { 0 \ } and the total quotient ring is the same as the field of fractions.
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Any nonzero element of the quotient ring will differ by a null sequence from such a sequence, and by taking pointwise inversion we can find a representative inverse element.
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From the point of view of abstract algebra, congruence modulo n is a congruence relation on the ring of integers, and arithmetic modulo n occurs on the corresponding quotient ring.
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The total quotient ring Q ( A \ times B ) of a product ring is the product of total quotient rings Q ( A ) \ times Q ( B ).
