<p>The aim of this paper is to develop a systematic B-Fredholm theory in semiprime Banach algebras. We first generalize Smyth’s important punctured neighbourhood theorem to B-Fredholm elements. Then using this result, we investigate the local spectral theory of B-Fredholm elements, including the localized left (resp. right) SVEP and a classification of components of the B-Fredholm resolvent set. Finally, in the context of semisimple Banach algebras, we characterize elements <i>f</i> such that <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="43037_2025_462_Article_IEq1.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="21" /> </InlineMediaObject> <EquationSource Format="TEX">\(f^{n}\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>f</mi> <mi>n</mi> </msup> </math></EquationSource> </InlineEquation> belongs to the socle for some <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="43037_2025_462_Article_IEq2.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="45" /> </InlineMediaObject> <EquationSource Format="TEX">\(n \in {\mathbb {N}}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>∈</mo> <mi mathvariant="double-struck">N</mi> </mrow> </math></EquationSource> </InlineEquation> from two different perspectives: one is the invariance of the B-Fredholm spectrum under commuting perturbation of such <i>f</i>, the other is <i>f</i> simultaneously being Riesz and B-Fredholm.</p>