Irrelevant ideal

Quality 0.00 · 0 views · Updated 14 days ago

In mathematics, the irrelevant ideal is the ideal of a graded ring generated by the homogeneous elements of degree greater than zero. It corresponds to the origin in the affine space, which cannot be mapped to a point in the projective space. More generally, a homogeneous ideal of a graded ring is called an irrelevant ideal if its radical contains the irrelevant ideal.

In mathematics, the irrelevant ideal is the ideal of a graded ring generated by the homogeneous elements of degree greater than zero. It corresponds to the origin in the affine space, which cannot be mapped to a point in the projective space. More generally, a homogeneous ideal of a graded ring is called an irrelevant ideal if its radical contains the irrelevant ideal.

The terminology arises from the connection with algebraic geometry. If R = k[x0, ..., xn] (a multivariate polynomial ring in n+1 variables over an algebraically closed field k) is graded with respect to degree, there is a bijective correspondence between projective algebraic sets in projective n-space over k and homogeneous, radical ideals of R not equal to the irrelevant ideal; this is known as the projective Nullstellensatz. More generally, for an arbitrary graded ring R, the Proj construction disregards all irrelevant ideals of R.

.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}body.skin-vector-2022 .mw-parser-output .reflist-columns-2{column-width:27em}body.skin-vector-2022 .mw-parser-output .reflist-columns-3{column-width:22.5em}.mw-parser-output .references[data-mw-group=upper-alpha]{list-style-type:upper-alpha}.mw-parser-output .references[data-mw-group=upper-roman]{list-style-type:upper-roman}.mw-parser-output .references[data-mw-group=lower-alpha]{list-style-type:lower-alpha}.mw-parser-output .references[data-mw-group=lower-greek]{list-style-type:lower-greek}.mw-parser-output .references[data-mw-group=lower-roman]{list-style-type:lower-roman}.mw-parser-output div.reflist-liststyle-upper-alpha .references{list-style-type:upper-alpha}.mw-parser-output div.reflist-liststyle-upper-roman .references{list-style-type:upper-roman}.mw-parser-output div.reflist-liststyle-lower-alpha .references{list-style-type:lower-alpha}.mw-parser-output div.reflist-liststyle-lower-greek .references{list-style-type:lower-greek}.mw-parser-output div.reflist-liststyle-lower-roman .references{list-style-type:lower-roman}

  • Sections 1.5 and 1.8 of .mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"""""""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#bf3c2c)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#bf3c2c)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}Eisenbud, David (1995), Commutative algebra with a view toward algebraic geometry, Graduate Texts in Mathematics, vol. 150, Berlin, New York: Springer-Verlag, ISBN 978-0-387-94269-8, MR 1322960
  • Hartshorne, Robin (1977), Algebraic Geometry, Graduate Texts in Mathematics, vol. 52, New York: Springer-Verlag, ISBN 978-0-387-90244-9, MR 0463157
  • Zariski, Oscar; Samuel, Pierre (1975), Commutative algebra volume II, Graduate Texts in Mathematics, vol. 29 (Reprint of the 1960 ed.), Berlin, New York: Springer-Verlag, ISBN 978-0-387-90171-8, MR 0389876

.mw-parser-output .asbox{position:relative;overflow:hidden}.mw-parser-output .asbox table{background:transparent}.mw-parser-output .asbox p{margin:0}.mw-parser-output .asbox p+p{margin-top:0.25em}.mw-parser-output .asbox-body{font-style:italic}.mw-parser-output .asbox-note{font-size:smaller}.mw-parser-output .asbox .navbar{position:absolute;top:-0.75em;right:1em;display:none}.mw-parser-output :not(p):not(.asbox)+style+.asbox,.mw-parser-output :not(p):not(.asbox)+link+.asbox{margin-top:3em}