<p>Let <i>R</i> be an associative ring with unit 1, and let <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(a, b, c\in R\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>,</mo> <mi>c</mi> <mo>∈</mo> <mi>R</mi> </mrow> </math></EquationSource> </InlineEquation> satisfy <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\((ac)^{2}a=abaca=acaba=a(ba)^{2}.\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mrow> <mo stretchy="false">(</mo> <mi>a</mi> <mi>c</mi> <mo stretchy="false">)</mo> </mrow> <mn>2</mn> </msup> <mi>a</mi> <mo>=</mo> <mi>a</mi> <mi>b</mi> <mi>a</mi> <mi>c</mi> <mi>a</mi> <mo>=</mo> <mi>a</mi> <mi>c</mi> <mi>a</mi> <mi>b</mi> <mi>a</mi> <mo>=</mo> <mi>a</mi> <msup> <mrow> <mo stretchy="false">(</mo> <mi>b</mi> <mi>a</mi> <mo stretchy="false">)</mo> </mrow> <mn>2</mn> </msup> <mo>.</mo> </mrow> </math></EquationSource> </InlineEquation> We prove that if <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(\alpha =1-ba\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>α</mi> <mo>=</mo> <mn>1</mn> <mo>-</mo> <mi>b</mi> <mi>a</mi> </mrow> </math></EquationSource> </InlineEquation> is generalized Drazin invertible, then <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(1-ac\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>1</mn> <mo>-</mo> <mi>a</mi> <mi>c</mi> </mrow> </math></EquationSource> </InlineEquation> is generalized Drazin invertible. This extends the results given by Chen and Abdolyousefi (Comm. Algebra, 49 (2021) 3263-3272) from Banach algebras to rings. Moreover, Jacobson’s lemma for generalized Fredholm elements relative to an ideal and Fredholm elements relative to a trace ideal is investigated in rings and in semisimple Banach algebras, respectively. Applying the above results, norm closure of hypercyclic operators is considered.</p>