10.3 Basic notions
The following is a list of basic notions in commutative algebra. Some of these notions are discussed in more detail in the text that follows and some are defined in the list, but others are considered basic and will not be defined. If you are not familiar with most of the italicized concepts, then we suggest looking at an introductory text on algebra before continuing.
R is a ring,
x\in R is nilpotent,
x\in R is a zerodivisor,
x\in R is a unit,
e \in R is an idempotent,
an idempotent e \in R is called trivial if e = 1 or e = 0,
\varphi : R_1 \to R_2 is a ring homomorphism,
\varphi : R_1 \to R_2 is of finite presentation, or R_2 is a finitely presented R_1-algebra, see Definition 10.6.1,
\varphi : R_1 \to R_2 is of finite type, or R_2 is a finite type R_1-algebra, see Definition 10.6.1,
\varphi : R_1 \to R_2 is finite, or R_2 is a finite R_1-algebra,
R is a (integral) domain,
R is reduced,
R is Noetherian,
R is a principal ideal domain or a PID,
R is a Euclidean domain,
R is a unique factorization domain or a UFD,
R is a discrete valuation ring or a dvr,
K is a field,
L/K is a field extension,
L/K is an algebraic field extension,
\{ t_ i\} _{i\in I} is a transcendence basis for L over K,
the transcendence degree \text{trdeg}(L/K) of L over K,
the field k is algebraically closed,
if L/K is algebraic, and \Omega /K an extension with \Omega algebraically closed, then there exists a ring map L \to \Omega extending the map on K,
I \subset R is an ideal,
I \subset R is radical,
if I is an ideal then we have its radical \sqrt{I},
I \subset R is nilpotent means that I^ n = 0 for some n \in \mathbf{N},
I \subset R is locally nilpotent means that every element of I is nilpotent,
\mathfrak p \subset R is a prime ideal,
if \mathfrak p \subset R is prime and if I, J \subset R are ideal, and if IJ\subset \mathfrak p, then I \subset \mathfrak p or J \subset \mathfrak p.
\mathfrak m \subset R is a maximal ideal,
any nonzero ring has a maximal ideal,
the Jacobson radical of R is \text{rad}(R) = \bigcap _{\mathfrak m \subset R} \mathfrak m the intersection of all the maximal ideals of R,
the ideal (T) generated by a subset T \subset R,
the quotient ring R/I,
an ideal I in the ring R is prime if and only if R/I is a domain,
an ideal I in the ring R is maximal if and only if the ring R/I is a field,
if \varphi : R_1 \to R_2 is a ring homomorphism, and if I \subset R_2 is an ideal, then \varphi ^{-1}(I) is an ideal of R_1,
if \varphi : R_1 \to R_2 is a ring homomorphism, and if I \subset R_1 is an ideal, then \varphi (I) \cdot R_2 (sometimes denoted I \cdot R_2, or IR_2) is the ideal of R_2 generated by \varphi (I),
if \varphi : R_1 \to R_2 is a ring homomorphism, and if \mathfrak p \subset R_2 is a prime ideal, then \varphi ^{-1}(\mathfrak p) is a prime ideal of R_1,
M is an R-module,
for m \in M the annihilator I = \{ f \in R \mid fm = 0\} of m in R,
N \subset M is an R-submodule,
M is an Noetherian R-module,
M is a finite R-module,
M is a finitely generated R-module,
M is a finitely presented R-module,
M is a free R-module,
if 0 \to K \to L \to M \to 0 is a short exact sequence of R-modules and K, M are free, then L is free,
if N \subset M \subset L are R-modules, then L/M = (L/N)/(M/N),
S is a multiplicative subset of R,
the localization R \to S^{-1}R of R,
if R is a ring and S is a multiplicative subset of R then S^{-1}R is the zero ring if and only if S contains 0,
if R is a ring and if the multiplicative subset S consists completely of nonzerodivisors, then R \to S^{-1}R is injective,
if \varphi : R_1 \to R_2 is a ring homomorphism, and S is a multiplicative subset of R_1, then \varphi (S) is a multiplicative subset of R_2,
if S, S' are multiplicative subsets of R, and if SS' denotes the set of products SS' = \{ r \in R \mid \exists s\in S, \exists s' \in S', r = ss'\} then SS' is a multiplicative subset of R,
if S, S' are multiplicative subsets of R, and if \overline{S} denotes the image of S in (S')^{-1}R, then (SS')^{-1}R = \overline{S}^{-1}((S')^{-1}R),
the localization S^{-1}M of the R-module M,
the functor M \mapsto S^{-1}M preserves injective maps, surjective maps, and exactness,
if S, S' are multiplicative subsets of R, and if M is an R-module, then (SS')^{-1}M = S^{-1}((S')^{-1}M),
if R is a ring, I an ideal of R, and S a multiplicative subset of R, then S^{-1}I is an ideal of S^{-1}R, and we have S^{-1}R/S^{-1}I = \overline{S}^{-1}(R/I), where \overline{S} is the image of S in R/I,
if R is a ring, and S a multiplicative subset of R, then any ideal I' of S^{-1}R is of the form S^{-1}I, where one can take I to be the inverse image of I' in R,
if R is a ring, M an R-module, and S a multiplicative subset of R, then any submodule N' of S^{-1}M is of the form S^{-1}N for some submodule N \subset M, where one can take N to be the inverse image of N' in M,
if S = \{ 1, f, f^2, \ldots \} then R_ f = S^{-1}R and M_ f = S^{-1}M,
if S = R \setminus \mathfrak p = \{ x\in R \mid x\not\in \mathfrak p\} for some prime ideal \mathfrak p, then it is customary to denote R_{\mathfrak p} = S^{-1}R and M_{\mathfrak p} = S^{-1}M,
a local ring is a ring with exactly one maximal ideal,
a semi-local ring is a ring with finitely many maximal ideals,
if \mathfrak p is a prime in R, then R_{\mathfrak p} is a local ring with maximal ideal \mathfrak p R_{\mathfrak p},
the residue field, denoted \kappa (\mathfrak p), of the prime \mathfrak p in the ring R is the field of fractions of the domain R/\mathfrak p; it is equal to R_\mathfrak p/\mathfrak pR_\mathfrak p = (R \setminus \mathfrak p)^{-1}R/\mathfrak p,
given R and M_1, M_2 the tensor product M_1 \otimes _ R M_2,
given matrices A and B in a ring R of sizes m \times n and n \times m we have \det (AB) = \sum \det (A_ S)\det ({}_ SB) in R where the sum is over subsets S \subset \{ 1, \ldots , n\} of size m and A_ S is the m \times m submatrix of A with columns corresponding to S and {}_ SB is the m \times m submatrix of B with rows corresponding to S,
etc.
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