Corrigendum to: on the inheritance of the strongly π-extending property

Abstract

In this corrigendum, we provide correct versions of Theorem 2.8 and Corollary 3.2, retract two consequences of the previous form of the Theorem 2.8. To this end, the main results of the article remain after the aforementioned changes.

Keywords:

• π-extending module
• projection invariant
• strongly π-extending module

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• Primary
• 16D10
• 16D50
• Secondary: 16D40

This article refers to:

On the inheritance of the strongly π-extending property

It has come to our attention that the statement of Theorem 2.8 in [3] is not true as it stands. We change the statement of this theorem and make a little modification in its proof. We also retract Corollary 2.15 and Theorem 3.1 that involve Theorem 2.8. Accordingly, we make suitable changes throughout the paper.

1. Page 3627, line 2 of the abstract, replace “In contrast to π-extending condition, it is shown that the former property is inherited by direct summands and Morita invariant” with “Like π-extending condition, it is shown that the former property is not inherited by direct summands”.

2. Page 3627, line 5 of the abstract, insert “nonsingular” before “free module”.

3. Page 3627, line -1, delete the sentence “We prove that a direct summand of a strongly π-extending module inherits the strongly π-extending property.”

4. Page 3628, line 6, delete the sentence “One more advantage of strongly π-extending condition is that it is a Morita invariant property (see, Theorem 3.1)”.

5. Page 3628, line 19, insert “when the” before “direct summands”.

6. Page 3630, line 17, replace “Next result shows that direct summands of strongly π-extending modules enjoy with the property” with “Observe that in [1, Example 5.5.] M_R is nonsingular module, and hence the direct summand D of M is not strongly π-extending by Proposition 2.5. Next result shows that when the direct summands of strongly π-extending modules enjoy with the property”.

7. Page 3630, replace Theorem 2.8 with the following version:

Theorem 2.8.

Let $M=M_1\oplus M_2$ be a strongly π-extending module. If M₂ is projection invariant in M, then both M₁ and M₂ are strongly π-extending.

Proof.

Let $S=En{d}_R(M)$. Then, there exists $e=e^2\in S$ such that $M=eM\oplus (1-e)M$, where $M_1=eM$ and $M_2=(1-e)M$. Note that $(1-e)M$ is projection invariant by hypothesis. Let $X{⊴}_p{M}_1=eM$. Since $X{⊴}_p{M}_1$ and $(1-e)M{⊴}_p\mathrm{\hspace{0.17em}M}$, we have $X\oplus (1-e)M$ is projection invariant submodule of M by [1, Lemma 4.13]. Then, there exists $f\in S_l(S)$ such that $X\oplus (1-e)M{\le }^{ess}fM$, where fM is a direct summand of M. ${(ef)}^2=\mathit{\text{efef}}=ef\in S$ since $f\in S_l(S)$. Thus, we can rewrite $M=efM\oplus (1-ef)M$. Note that $efM\le eM$. Then $eM=efM\oplus (eM\cap (1-ef)M)$ by modular law.

Now, we claim that $X{\le }^{ess}efM$ and efM is a fully invariant direct summand of eM. Since $f\in S_l(S)$ and ${(ef)}^2=ef\in S,efM\subseteq eM\cap fM$. Let $x\in eM\cap fM$. Then there exist m₁, m₂ such that $x=e{m}_1=f{m}_2$. This implies that $ex=e{m}_1=ef{m}_2\in efM$. Thus $efM=eM\cap fM$ holds. As $X\oplus (1-e)M{\le }^{ess}fM$, we also have $eM\cap (X\oplus (1-e)M){\le }^{ess}eM\cap fM=efM$. Since $X\le eM$, we get $eM\cap (X\oplus (1-e)M)=X$ by modular law. Therefore, $X{\le }^{ess}efM$. Next, we show that efM is fully invariant in eM. It is clear that $En{d}_R(eM)=eSe$. Now, $eSe(efM)=e(Se(fM))\subseteq efM$ since fM is fully invariant in M. Therefore, efM is a fully invariant direct summand of $eM$. Hence M₁ is strongly π-extending.

Let Y be a projection invariant submodule of M₂. Since $M_2{⊴}_{p}M,Y{⊴}_{p}M$. Then, there exists a fully invariant direct summand K of M such that $Y{\le }^{ess}K$. It follows that $Y{\le }^{ess}K\cap M_2$. Note that $M=K\oplus K\prime$ for some $K\prime \le M$. Since $M_2{⊴}_{p}M$, we can write $M_2=(K\cap M_2)\oplus (K\prime \cap M_2)$. However, K is fully invariant in M, so it is projection invariant in M. Hence $K=(K\cap M_1)\oplus (K\cap M_2)$, where $K\cap M_i{⊴}_p{M}_i$ for i = 1, 2. Therefore, $K\cap M_2$ is a fully invariant direct summand of M₂ which yields that M₂ is strongly π-extending.

• 8. Page 3630, lines 1-2 of Proposition 2.9, delete “(nonsingular π-extending, respectively)” and “(strongly π-extending, respectively)”.

• 9. Page 3630, in the proof Proposition 2.9, delete “If M is nonsingular π-extending, then by Propositions 2.3, 2.5 and Theorem 2.8, X is strongly π-extending.”

• 10. Page 3632, line 6, replace “we are able to show that the strongly π-extending property is a Morita invariant unlike the case for the π-extending. We apply the former result to the endomorphism ring of a free module” with “we apply the former result to the endomorphism ring of a nonsingular free module”.

• 11. Page 3632, replace Corollary 3.2 with the following version:

Corollary 3.2.

Let R be a right nonsingular strongly π-extending ring and F_R be a free right R-module. Then $En{d}_R(F)$ is right strongly π-extending.

Proof.

By Theorem 2.14, F_R is strongly π-extending, so it is π-extending. Thus $En{d}_R(F)$ is right π-extending from [2, Theorem 16]. Hence $En{d}_R(F)$ is strongly π-extending by Proposition 2.5.

12. Page 3632, line 1 of the proof of Theorem 3.3, replace “Let $e_{22}=\left[\begin{array}{cc}0& 0\\ 0& 1\end{array}\right]\in T$. Then, $e_{22}T$ is also right strongly π-extending by Theorem 2.8.” with “Let $e=\left[\begin{array}{cc}1& 0\\ 0& 0\end{array}\right]\in T$. Then, $eT=\left[\begin{array}{cc}1& 0\\ 0& 0\end{array}\right]\left[\begin{array}{cc}S& M\\ 0& R\end{array}\right]=\left[\begin{array}{cc}S& M\\ 0& 0\end{array}\right]$ is an ideal of T. Hence it is projection invariant in T. Thus $\left[\begin{array}{cc}0& 0\\ 0& R\end{array}\right]$ is right strongly π-extending by Theorem 2.8. Hence R_R is right strongly π-extending.”

Acknowledgment

The authors thank to Fatih Karabacak and Özgür Taşdemir for their careful reading of the paper and for drawing to their attention the fact that Theorem 2.8 seems to be not true.