Corrigendum

“Bilinear products with A.C.C. on annihilators” by Gustavo Mezzetti *, Communications in Algebra^® , 30 (2), 2002, pp. 1039–1047.

Table

Place	Errata	Corrige
p. 1041, 1st display	ℱ(X) = {W ∈ ℒ¯(N) etc.	ℱ⊥(X) = {W ∈ ℒ¯(N) etc.
p. 1041, l. 9	Clearly, X ^⊥ ∈ ℱ(X), so ℱ(X) ≠ Ø has maximal elements.	Clearly, X ^⊥ ∈ ℱ⊥(X), so ℱ⊥(X) ≠ Ø has maximal elements
p. 1041, l. 10	= (X′)^⊥ ∈ F(X)	= (X′)^⊥ ∈ ℱ⊥(X)
p. 1041, l. 11	therefore, W/Xis of finite length	therefore, W/X ^⊥is of finite length
p. 1041, l. 13	Moreover, ℱ(X) is filtered; indeed let W 1, W 2 ∈ ℱ(X), say W i  = X i	Moreover, ℱ⊥(X) is filtered; indeed let W 1, W 2 ∈ ℱ⊥(X), say W i  =
p. 1041, 2nd display	W 1 + W 2 = X 1 + X 2  ≤ (X 1 ∩ X 2)	W 1 + W 2 =  +   ≤ (X 1 ∩ X 2)^⊥
p. 1041, l. -10	we conclude that (X 1 ∩ X 2) is an element of ℱ(X)	we conclude that (X 1 ∩ X 2)^⊥is an element of ℱ⊥(X)
p. 1041, l. -9	Thus we see that ℱ(X) has…	Thus we see that ℱ⊥(X) has…
p. 1041, l. -8	by F(X). Note … that F(X)/Xis of finite length.	by F ⊥(X). Note … that F ⊥(X)/X ^⊥is of finite length.
p. 1041, l. -3	finite length of N/X	finite length of N/X ^⊥
p. 1041, l. -2	contained in F(X)/X, so that F(X)/ is…	contained in F ⊥(X)/X ^⊥, so that F ⊥(X)/X ^⊥is…
p. 1041, last line	N/X	N/X ^⊥
p. 1042, l. 1	Let L ≤ N/Xbe … write L = W/Xwith X ≤ W ≤ N.	Let L ≤ N/X ^⊥be … write L = W/X ^⊥with X ^⊥ ≤ W ≤ N.
p. 1042, 1st display	X × N/X → U	X × N/X ^⊥ → U
p. 1042, 2nd display	Ann N/X (Ann X (L)) =  .	Ann N/X ^⊥ (Ann X (L)) =  .
p. 1042, l. 5	Write F(X) = X 0with…	Write F ⊥(X) = with…
p. 1042, 3rd display	F(X) = X 0  ≤ (W ∩ X) + X 0  ≤ (W ∩ X ∩ X 0)  = (W ∩ X 0)	F ⊥(X) =   ≤ (W ^⊥ ∩ X)^⊥ +   ≤ (W ^⊥ ∩ X ∩ X 0)^⊥  = (W ^⊥ ∩ X 0)^⊥
p. 1042, l. 6	But now, (W ∩ X 0) ∈ ℱ(X):	But now, (W ^⊥ ∩ X 0)^⊥ ∈ ℱ⊥(X):
p. 1042, 4th display	↪  ⊕ ,	↪  ⊕ ,
p. 1042, l. 7	it suffices to show that is…	it suffices to show that is…
p. 1042, 5th display
p. 1042, l. 9	maximality of F(X)	maximality of F ⊥(X)
p. 1042, 6th display
p. 1042, l. -4	Mhas the A.C.C. on subsets of N	Mhas the A.C.C. on annihilators of subsets of N
p. 1042, l. -2	X = Z	X ^⊥ = Z ^⊥
p. 1043, 1st display	N/X = N/Z ↪ Z ↪  (R ^n) = (R)^n = U ^n	N/X ^⊥ = N/Z ^⊥ ↪  Z* ↪ (R ^n)* = (R*)^n = U ^n
p. 1043, l. 2	N/Xhas essential…	N/X ^⊥has essential…
p. 1043, l. 9	then F(M)Nand N/F(M) ≠ 0.	then F ^⊥(M) ⪇ Nand N/F ⊥(M) ≠ 0.
p. 1043, l. 10	N/F(X) has a simple…	N/F ⊥(X) has a simple…
p. 1043, l. 11	the top of F(M)	the top of F ⊥(M)
p. 1043, 2nd display	V = Hom R (V, U) = …	V* = Hom R (V,U) = …
p. 1043, l. -10	Vhas finite length	V* has finite length
p. 1044, ll. 10–11	R ⟨end of line⟩U	R U
p. 1045, 1st display	K ↦ K	K ↦ K ^⊥⊥
p. 1045, 2nd display	L ↦ L	L ↦ L ^⊥⊥
p. 1045, l. 16	(where· denotes the annihilator…)	(where ·^⊥denotes the annihilator…)
p. 1045, 3rd display	R M → N	R M → N*
p. 1045, 4th display	N T  → M	N T  → M*
p. 1046, l. 13	mynsymmetric	minsymmetric

Notes

*Correspondence: G. Mezzetti. E-mail: [email protected].