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Complete intersection ring
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<p>In <a href="/facts/Commutative_algebra/mKtUdPCT">commutative algebra</a>, a complete intersection ring is a <a href="/facts/Commutative_ring/QS1r2ovR">commutative ring</a> similar to the <a href="/facts/Coordinate_ring/WVynz2Kn">coordinate rings</a> of varieties that are <a href="/facts/Complete_intersection/UoFKxRqC">complete intersections</a>. Informally, they can be thought of roughly as the <a href="/facts/Local_ring/gRTdg5Qx">local rings</a> that can be defined using the "minimum possible" number of relations. </p><p>For Noetherian local rings, there is the following chain of inclusions: </p> <a href="/facts/Universally_catenary_ring/5gK61uEy">Universally catenary rings</a> ⊃ <a href="/facts/Cohen%25E2%2580%2593Macaulay_ring/4G8wymtC">Cohen–Macaulay rings</a> ⊃ <a href="/facts/Gorenstein_ring/D9bw8ocr">Gorenstein rings</a> ⊃ complete intersection rings ⊃ <a href="/facts/Regular_local_ring/P9s5Fvrl">regular local rings</a>