Quotienting is a generalisation of division

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SUMMARY

Quotients in abstract algebra represent a generalization of division, extending the concept from single elements to sets. This discussion highlights that linear combinations serve as a generalization of multiples, establishing a framework for determining when an element is a linear combination of others. In a commutative setting, an object is considered zero if it is a linear combination of specified objects, emphasizing the foundational role of quotients in algebraic structures.

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Quotients in abstract algebra is a generalisation of division. While the latter works only for single elements, the former is for dividing sets. Amazing stuff!
 
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linear combinations are a generalization of multiples, so asking when an element is a linear combination of other elements is a generalization of when it is a multiple of another element, yes.

(a quotient object of course is one where an object is considered zero iff it is a linear combination of given objects, at least in a commutative setting.)
 

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