MATH 436 Notes: Examples of Rings
نویسنده
چکیده
i∈N R = {(a0, a1, a2, . . . )|aj ∈ R} is a countable direct product of (R,+) with itself. Thus the elements of R[[x]] are arbitrary sequences with entries in R where addition is componentwise. We define a multiplication ⋆ via (a0, a1, . . . ) ⋆ (b0, b1, . . . ) = (c0, c1, . . . ) where cn = ∑n k=0 akbn−k for all n ∈ N. It is easy to check that the sequence (1, 0, 0, . . . ) is a two-sided identity for this multiplication. It is also easy to check that the Mth term for the sequence given by ((a0, a1, . . . ) ⋆ (b0, b1, . . . ) ⋆ (c0, c1, . . . )) is ∑M n=0( ∑n k=0 akbn−k)cM−n. Since R is a ring, this equals ∑k+l+s=M k,l,s≥0 akblcs. This final expression is also the Mth term of the sequence (a0, a1, . . . ) ⋆ ((b0, b1, . . . ) ⋆ (c0, c1, . . . )) and so we see that ⋆ makes R[[x]] into a monoid. Finally since R has distributivity, it is easy to check that R[[x]] also does. Thus (R[[x]],+, ⋆) is a ring. It is commutative if and only if R is. We will now introduce a formal variable symbol x and adopt the more intuitive notation of ∑∞ n=0 anx n for (a0, a1, a2, . . . ). (Note: Changing the symbol of the index of summation does not change the element represented.) The addition and multiplication defined above now can be expressed as:
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