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The PDF does not include interactive content such as simulations, videos, and quizzes and is not vetted for accessibility. pdf) or view presentation slides online. In ordinary algebra, there is a thin line dividing the theory of commutative rings from the theory of associative rings: a commutative ring Ris just an associative ring whose elements satisfy the additional identity xy= yx. 1 2=3 Although it is a stretch in this case, suppose that one does not know one of the numbers. 3 There’s a lesson here: domains and codomains are important, and changing them really changes the properties of a function! So when you define or specify a function, it really does matter that you say what its domain and codomain are. The following is a list of the most cited articles based on citations published in the last three years, according to CrossRef. Equivalence Relations 29 5. 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