By Huishi Li
Designed for a one-semester path in arithmetic, this textbook provides a concise and useful creation to commutative algebra when it comes to basic (normalized) constitution. It exhibits how the character of commutative algebra has been utilized by either quantity idea and algebraic geometry. Many labored examples and a couple of challenge (with tricks) are available within the quantity. it's also a handy reference for researchers who use easy commutative algebra.
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Extra info for An Introduction to Commutative Algebra: From the Viewpoint of Normalization
2 7, follow ~ ~ '. I. , s, . Example (i) The symmetric polynomial Preliminaries 31 is written lexicographically. And by the method given in the proof we may derive that f = s l s 2 + 3 s 3 . Similarly, ( I C 1 + 2 2 ) ( z l + x 3 ) ( 2 2 + 2 3 ) = s 1 s 2 - s 3 . An application of symmetric polynomials to field extension is given as follows. If K C L is a field extension, a,xn a , - ~ x " - ~ . ,T~ E L , then, f(x) factors in L [ x ]as + f(x) = a,(z - T1)(Z - T Z ) . + = an(xn where ci = ( - ~ ) Z S ~ ( T ~ , T ~ , of both sides, we have +.
X , over a ring A. +a,=i Ax:' . . xzn , i E N,which is called the ith homogeneous part of R. Show that, as A-modules, R = @ i E ~ R and i, that, as subsets, RiRj = Ri+j, i , j E N.
Hint: Note that ()(A), c K . Any Q-linear ring monomorphism u: K C induces a Q-linear ring monomorphism GI: Q1 and a Q-linear ring monomorphism 0 2 : Q1, such that ~ ( 6=) a l ( 4 ) 02(&). ) Can you generalize this result to 6 = fi 4 for arbitrary squarefree p # q? + a(&) + a, a(&) Q(a) --f -+ + a(&) -+ + + Commutative Algebra 40 6. Free Abelian Groups of Finite Rank + Let G be an abelian group with the binary additive operation and the identity element 0. For g E G , we write Z g for the cyclic subgroup of G generated by g , and consequently, we write C g E n Z g for the subgroup of G generated by a nonempty subset R C G.