Corrigendum to: Goldie Extending Modules

Abstract

This corrigendum is written to correct an error in Corollary 2.5(ii) and an error in the proof of the converse of Theorem 2.7 of Akalan, Birkenmeier, and Tercan [1].

Key Words:

• Goldie extending module

2000 Mathematics Subject Classification:

• Primary 16D70
• Secondary 16D50, 16D40

In Corollary 2.5 delete part (ii) and its proof.

In Example 2.6(iii), replace “in Corollary 2.5(ii) \ldots right self-injective.” with “gives a subdirectly irreducible commutative ring. By Corollary 2.5(i), S is right self-ejective but not right self-injective even though M is injective as an R-module in this split-null extension.”

Lines 4–6 of the proof of the converse of Theorem 2.7 on p. 670 of Akalan et al. [1] are in error. To correct this error, replace “H′ ∩ L…. Therefore […]” with the following:

“H′ ∩ H ≤ ^ess H. Let K = H′ ∩ H ∩ L. There exists C ≤ L such that K ∩ C = 0 and K ⊕ C ≤ ^ess L. Let B = {b ∈ C | −g(b) + b ∈ H′}. Observe that B ≤ C. We claim that B ≤ ^ess C. So let 0 ≠ c ∈ C. Then −g(c) + c ∈ H. If −g(c) + c = 0, then c ∈ M ₁ ∩ L = 0, a contradiction. Thus −g(c) + c ≠ 0. Then there exists r ∈ R such that 0 ≠ (−g(c) + c)r = − g(cr) + cr ∈ H′ ∩ H. So 0 ≠ cr ∈ B. Hence B ≤ ^ess C.

Observe that K ⊕ B ≤ ^ess L. Now let k + b ∈ K ⊕ B, where k ∈ K and b ∈ B. Suppose that π: M → M ₁ is the projection along H′ (i.e., ker π =H′). Then π(k + b) = π(b) = π(g(b) − g(b) + b) = π(g(b)) + π(−g(b) + b) = π(g(b)) = g(b). Recall that k ∈ H ∩ L. Then there is a y ∈ L such that k = − g(y) + y. Hence g(y) = y − k ∈ L ∩ M ₁ = 0. So y = k and 0 = g(y) = g(k). Then π(k + b) = g(b) = g(k + b). Therefore […].”

In the proof of Theorem 3.1(ii), line 3, replace “Lemma 1.5” with “Lemma 7.5.”

In the proof of Theorem 3.1(ii), line 4, replace the last subscript “M” with “M′.”

In the proof of Corollary 3.10, line 2, replace “7.2” with “6.7.2.”

In Corollary 5.2(iii), line 3, replace “S on” with “submodule S of.”

ACKNOWLEDGMENT

We are grateful to Jae Keol Park and S. Tariq Rizvi for providing a counterexample to Corollary 2.5(ii) and to Yongduo Wang who questioned the proof of the converse of Theorem 2.7.

Notes

Communicated by T. Albu.