Math 557 Final exam study guide (University of Idaho, Fall 2022)

Brooks Roberts

The final exam will cover the material presented in class from chapters 1, 2, 3, 4, 5, 6, 7, 8, and 10 of ourtext, and will take place on Friday, December 16, 2022 from 10:15 AM to 12:15 AM. For the exam youshould:
be able to define:• free R-module;• integral domain;• rank(F ) for F free R-module;• field;• Noetherian R-module;• Euclidean domain;• Artinian R-module;• principal ideal domain (PID);• Artinian ring;• unique factorization domain (UFD);• simple module;• Noetherian ring;• unit in a ring;• composition series;• composition factor;• zero divisor in a ring;• the length `(M ) of a composition series for M ;• nilpotent element of a ring;• Jacobson radical Jac(R);• irreducible element in an integral domain;• prime element in an integral domain;• comaximal ideals;• radical of an ideal;• nilradical of R;• ideal quotient (colon ideal);• maximal ideal;• prime ideal;• primary ideal;• decomposable ideal;• irreducible ideal;

be able to state:• Chinese remainder theorem (see the online lecture notes for the statement);• First uniqueness theorem for primary decomposition;• Second uniqueness theorem for primary decomposition;....................................................
• First, second, and third isomorphism theorem
for R-modules;• minimal prime ideal of I;• Jordan-Hölder theorem;• embedded prime ideal of I;• Hilbert basis theorem;• minimal primary decomposition;• Nakayama’s lemma;• Spec(R);• Krull’s intersection theorem;• Var(I);• Elementary divisors theorem (see online lecture• Min(I);• assR (I);....................................................

notes);• Structure theorem for finitely generated modules over a PID (see online lecture notes);• multiplicatively closed set;
• the ring of fractions S −1 R;• RP if P is a prime ideal of R;• quasi-local ring;understand and know:• The natural map surjective map R → R/I thathas kernel I;• quasi-semi-local ring;• The isomorphism theorem;• local ring;• Given an ideal I of R, the bijection between the• semi-local ring;ideals J such that I ⊆ J and the ideals of R/I;

• R-module;• Given a ring homomorphism f : R → S, the• Ann(M );contraction and extension maps on ideals de-• short exact sequence;termined by f , especially when f is surjective;

Math 557 Final exam study guide (University of Idaho, Fall 2022)• How to characterize an ideal I of R being maximal, prime, or primary in terms of the quotient

Brooks Roberts• M is Noetherian if and only if every submoduleof M is finitely generated;• For an R-module M with submodule N :R/I;• The equivalent conditions for a non-zero ideal

M is Noetherian (Artinian)(r) in a PID to be a prime ideal;• If R is an Euclidean domain, then R is a PID;
• If R is a PID, then R is a UFD;m
N and M/N are Noetherian (Artinian);• If K is a field, K[X] is Euclidean, hence a PID,hence a UFD;

• For R-modules M1 , . . . , Mn ,• If R is a UFD, then R[X] is a UFD;nM

• Every proper ideal is included in a maximalideal;√TT• I = P ∈Var(I) P = P ∈Min(I) P ;• Formulas involving radicals:p√√√
IJ = (I ∩ J) = I ∩ J;
√P n = P if P is prime;√• Q maximal =⇒ Q primary;n• M maximal =⇒ M primary;• What are the primary ideals in a PID;
• If R is Noetherian, then every proper ideal hasa primary decomposition;

Mi Noetherian (Artinian)i=1m
M1 , . . . , Mn Noetherian (Artinian);• If R is Noetherian (Artinian), then every
finitely generated R-module is Noetherian (Artinian);• N is a simple R-module if and only if N ∼= R/Mwith M a maximal ideal;• If R is a PID and not a field, and M is an R-....................................................

module, then M has finite length if and only if• The universal property of the natural map R →M is finitely generated and there exists r ∈ R,
r 6= 0, such that rM = 0.

S −1 R;• If P is a prime ideal of R, then S = R − P is a• If R is a non-zero Noetherian ring, and N is anmultiplicatively closed set, and S −1 R = RP isR-module, then N has finite length if and only if

quasi-local;

N is finitely generated and there exists maximal• For the natural map f : R → S −1 R,(J c )e = J;√√( I)e = I e ;ideals M1 , . . . , Mn such that M1 · · · Mn · N = 0;
• If f : R → R0 is a surjective ring homomorphism and R is Noetherian, then R0 is Noethe-I ∩ S 6= ∅ ⇐⇒ I e = S −1 R;• There are bijections:e,c{P ∈ Spec(R) : P ∩ S = ∅}  Spec(S −1 R),e,c{Q primary, Q ∩ S = ∅}  {Q0 ⊆ S −1 R primary};• If R is Noetherian, then S −1 R is Noetherian;

rian;• If I is a finitely generated ideal of R, then there√
exists n ∈ N such that ( I)n ⊆ I;
• R is Artinian if and only if R is Noetherian andevery prime ideal of R is maximal;• If R is Artinian then R is semi-local;• Given an R-module M with submodule N , theand be able to demonstrate knowledge of the above
concepts and results in the context of simple prob-bijection between submodules N 0 of M suchlems and questions for the examples Z, Z/N Z, K[X],

that N ⊆ N 0 and submodules of M/N ;and K[X1 , . . . , Xn ].

• External and internal direct sums;• Every R-module is the quotient of free Rmodule;