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As an application of our techniques we\u00a0demonstrate that each extension of groupoids <jats:inline-formula>\n              <jats:alternatives>\n                <jats:tex-math>$${\\mathcal {N}}\\rightarrow {\\mathcal {E}}\\rightarrow {\\mathcal {G}}$$<\/jats:tex-math>\n                <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>N<\/mml:mi>\n                    <mml:mo>\u2192<\/mml:mo>\n                    <mml:mi>E<\/mml:mi>\n                    <mml:mo>\u2192<\/mml:mo>\n                    <mml:mi>G<\/mml:mi>\n                  <\/mml:mrow>\n                <\/mml:math>\n              <\/jats:alternatives>\n            <\/jats:inline-formula> gives rise to a groupoid crossed\u00a0product of <jats:inline-formula>\n              <jats:alternatives>\n                <jats:tex-math>$${\\mathcal {G}}$$<\/jats:tex-math>\n                <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mi>G<\/mml:mi>\n                <\/mml:math>\n              <\/jats:alternatives>\n            <\/jats:inline-formula> by the groupoid ring of <jats:inline-formula>\n              <jats:alternatives>\n                <jats:tex-math>$${\\mathcal {N}}$$<\/jats:tex-math>\n                <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mi>N<\/mml:mi>\n                <\/mml:math>\n              <\/jats:alternatives>\n            <\/jats:inline-formula> which recovers the groupoid ring of\u00a0<jats:inline-formula>\n              <jats:alternatives>\n                <jats:tex-math>$${\\mathcal {E}}$$<\/jats:tex-math>\n                <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mi>E<\/mml:mi>\n                <\/mml:math>\n              <\/jats:alternatives>\n            <\/jats:inline-formula> up to isomorphism. Furthermore, we make the somewhat surprising observation that our classification methods naturally transfer to the class of groupoid crossed products, thus providing a classification theory for this class of rings. Our study is motivated by the search for natural examples of groupoid crossed products.<\/jats:p>","DOI":"10.1007\/s10485-024-09795-8","type":"journal-article","created":{"date-parts":[[2024,12,16]],"date-time":"2024-12-16T09:04:21Z","timestamp":1734339861000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Non-Abelian Extensions of Groupoids and Their Groupoid Rings"],"prefix":"10.1007","volume":"33","author":[{"given":"Nat\u00e3","family":"Machado","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Johan","family":"\u00d6inert","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Stefan","family":"Wagner","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2024,12,16]]},"reference":[{"key":"9795_CR1","doi-asserted-by":"publisher","DOI":"10.1016\/j.geomphys.2022.104640","volume":"180","author":"P Antonini","year":"2022","unstructured":"Antonini, P., Guido, D., Isola, T., Rubin, A.: A note on twisted crossed products and spectral triples. 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