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In [[mathematics]], in the subfield of [[ring theory]], a ring ''R'' is a '''polynomial identity ring''' if there is, for some ''N'' > 0, an element ''P'' other than 0 of the [[free algebra]], Z<''X''<sub>1</sub>, ''X''<sub>2</sub>, ..., ''X''<sub>''N''</sub>>, over the [[ring of integers]] in ''N'' variables ''X''<sub>1</sub>, ''X''<sub>2</sub>, ..., ''X''<sub>''N''</sub> such that for all ''N''-[[tuple]]s ''r''<sub>1</sub>, ''r''<sub>2</sub>, ..., ''r''<sub>''N''</sub> taken from ''R'' it happens that
<br><br>In my humble opinion, Case provides among the most exquisite handles on pocket knives. For instance, my father gave me a Peanut pocket knife for Christmas this year. It’s handles are made from MAMMOTH TOOTH. That’s proper, MAMMOTH TOOTH. It is a attractive, one in every of a form piece that [http://Sss.Chaoslab.ru/tracker/mim_plugs/newticket?reporter=anonymous&summary=AttributeError%3A+%27Environment%27+object+has+no+attribute+%27get_db_cnx%27&description=%0D%0A%0D%0AShould+you%27re+heading+down+the+path+of+an+aggressive+DIY+life-style+and+really+feel+that+the+flexibility+to+store+a+number+of+blades+on+a+knife+is+crucial%2C+we+suggest+that+you+just+go+together+with+the+Olympia+Turbopro+%28%24sixteen%29.+Even+with+its+extremely+compact+physique+%28thinner+than+the+Fastback+II%29%2C+it+nonetheless+has+the+potential+to+deal+with+5+further+blades.+It+also+has+an+auto-load+characteristic+to+make+blade+modifications+freakishly+simple.+It%27s+a+good+sturdy+knife+and+the+butt+finish+has+a+small+carabiner+clip+so+it+can+be+hooked+on+a+%5Bhttps%3A%2F%2Fwww.google.com%2Fsearch%3Fhl%3Den%26gl%3Dus%26tbm%3Dnws%26q%3Dbelt%2Bloop%26btnI%3Dlucky+belt+loop%5D.%0D%0A%0D%0ABecause+of+the+variety+of+stored+blades+and+the+comfortable+rubber+padding+on+the+grip+area%2C+that+is+going+to+be+a+winner+for+someone+who+actually+has+a+knife+of+their+fingers+all+day+lengthy.+Folding+knives+are+extra+compact+than+retractables.+The+standard+retractable+knife+is+about+seven+inches+long%2C+in+order+that+they%27re+a+little+bit+much+for+the+pants+pocket+and+do+not+include+belt+clips.+But+because+a+folding+knife+is+hinged+on+the+center%2C+when+they%27re+in+storage-mode%2C+they%27re+only+about+three+to+five+inches+long%3B+when+unfolded%2C+they+provide+the+same+deal+with+length+to+the+retractables.+All+worldwide+orders+should+have+a+ship-to+destination+exterior+of+the+United+States.+We+can%27t+help+prospects+with+worldwide+billing+addresses+delivery+to+U.S.+addresses.%0D%0A%0D%0AThere+may+be+normally+a+variety+of+dialogue+surrounding+the+topic+of+serrated+steak+knives+and+non-serrated+steak+knives.+Serrated+Steak+Knives+assist+you+to+simply+lower+via+meats+with+a+%E2%80%9Csawing%E2%80%9D+motion+whereas+non-serrated+steak+knives+allow+for+a+cleaner+cut.+At+Rada+Cutlery+we+offer+a+Serrated+Steak+Knife+and+a+Utility%2FSteak+Knife%3B+which+is+non-serrated.+They+are+each+constituted+of+a+excessive+carbon+chrome+steel+and+both+easily+lower+via+meat.+So+when+it+comes+down+to+picking+between+serrated+and+non-serrated+it%27s+based+on+desire+and+whether+or+not+you+need+a+clear+minimize.%0D%0A%0D%0AA+rescue+knife+will+probably+be+used+for+slicing+harness+from+persons+trapped+inside+a+car%2Fairplane%2Fyou+title+it+after+an+accident+and+in+addition+for+other+issues%2C+like+slicing+open+a+fuselage.+Portability+is+necessary+in+such+circumstances%3B+therefore+you+have+to+choose+fastidiously+the+scale%2C+weight+and+bulkiness+of+your+knife.+A+big%2C+heavy+knife+is+difficult+to+carry+around%2C+changing+into+a+liability+as+a+substitute+of+an+asset.+The+cutting+edge+is+the+underside+edge+of+the+blade+that+runs+from+the+heel+to+the+tip+of+the+blade.+It+is+extremely+sharp+and+might+be+straight+minimize+or+serrated.+The+innovative+is+used+to+slice%2C+lower+or+chop+meals+gadgets+each+giant+and+small%2C+with+the+center+of+the+blade+getting+used+most+often.+A+simple%2C+cheap+box+cutter+can+save+the+blade+on+a+more+expensive+survival+knife.+Mini+Santoku+Knife%0D%0A%0D%0AA+more+useful+characteristic+for+the+casual+consumer+is+the+%E2%80%9Cgut+hook%2C%E2%80%9D+which+helps+you+to+lower+string+and+different+thin+objects+with+out+opening+the+blade.+It+is+a+deep+lower-away+at+the+back+of+the+handle+that+exposes+a+small+portion+of+the+blade+when+the+knife+is+in+the+closed+position+%28don%27t+worry%2C+it%27s+practically+unattainable+to+chop+yourself+on+it%29.+This+lets+you+cut+a+size+of+twine+or+open+a+bag+of+cat+kibble+whereas+retaining+the+blade+safely+tucked+into+the+tool+A+look+at+the+spring-loaded+blade+cartridge+within+the+Milwaukee+Self-Retracting+Knife+It%27s+worth+noting+that+there+are+a+variety+of+Amazon+opinions+that+describe+the+blade+releasing+from+the+software+throughout+use.+Comparable+comments+appear+with+this+Stanley+knife+Consumer+opinion+%28ratings%2Fopinions%2C+as+much+as+5+stars%29.%0D%0A%0D%0AEnsure+knives+are+kept+sharp.+A+dull+knife+requires+extra+effort+when+utilizing+and+the+extra+effort+that+is+used%2C+the+more+likelihood+the+blade+can+slip+and+cause+injury.+To+test+the+sharpness+of+a+blade%2C+pull+the+blade+evenly+over+a+tomato.+The+blade+should+make+a+clean+lower+through+the+pores+and+skin+with+little+effort.+Another+technique+of+testing+the+sharpness+of+a+blade+is+to+take+a+sheet+of+paper+and+try+to+cut+strips+from+it+by+slicing+with+the+knife.+The+knife+ought+to+cut+easily+through+the+paper+with+out+resistance.%0D%0A%0D%0AIn+actual+fact%2C+we+found+only+a+few+extra+knives+in+any+of+the+units+that+weren%27t+rendered+redundant+by+certainly+one+of+these+core+knives.+Though+the+pointed-tipped+carving+knives+included+in+two+of+the+units+have+been+completely+practical%2C+neither+one+did+a+better+job+reducing+up+roast+beef+or+roast+chicken+than+the+chef%27s+knife+already+in+these+units.+It+was+the+identical+story+with+the+5%C2%BD-+to+7-inch+santoku+or+nakiri+blades+that+came+in+most+of+the+sets.+These+Asian-inspired+knives+have+been+good+to+have+round+for+vegetable+prep%2C+however+did+they+do+something+the+chef%27s+knife+could+not%3F+Probably+not.%0D%0A%0D%0Aonerous+to+take+aside%2C+blade+change%2Cidentical+as+all+different+stanley+knifes.+you+lengthen+the+blade+to+the+max+and+press+the+buttom+to+the+left+of+the+tip+of+the+knife+and+pull+it+out%2C+flip+the+blade+or+change+it+Simon+Medhurst%2C+President+of+Fast+Tools+%C2%AE+%2C+talking+concerning+the+Rapid+Edge+%C2%AE+innovation%2C+%E2%80%9CDue+to+the+serration%2C+the+blade+makes+use+of+three+to+5+reducing+surfaces+as+a+substitute+of+the+only+chopping+surface+found+on+a+standard+blade.+This+configuration+implies+that+the+blade+stays+sharper+longer+and+cuts+straighter.+In+case+you+loved+this+post+and+you+would+want+to+receive+details+with+regards+to+%5Bhttp%3A%2F%2Fwww.thebestpocketknifereviews.com%2Fbest-utility-knife-top-rated-knives-analysis%2F+craftsman+utility+Knife%5D+generously+check+out+our+web-site.+%E2%80%9D+I+noticed+that+gerber+when+it+came+out+and+an+Arfcommer+reviewed+it+so+I+purchased+one.+If+you+have+to+put+some+drive+into+your+minimize+it+will+let+go+of+the+blade.+I+ended+up+shedding+it+and+don%27t+miss+it.+Posted+by+Sarah+Dunn+on+twenty+seventh+Jul+2014+Link+to+purchase+stated+knife&create=Create I completely] love. The Backpocket knife Case despatched [http://Www.Thebestpocketknifereviews.com/best-utility-knife-top-rated-knives-analysis/ http://Www.Thebestpocketknifereviews.com/best-Utility-knife-top-rated-Knives-analysis] us has an attractive Chestnut bone handle. The colour is rich and the jigging (crevices in the bone) provide good grip in moist situations. The ideas can break off or the blade chip if a ceramic blade Pocket knife is [http://sss.Chaoslab.ru/tracker/mim_plugs/newticket?reporter=anonymous&summary=AttributeError%3A+%27Environment%27+object+has+no+attribute+%27get_db_cnx%27&description=Thanks+for+taking+curiosity+in+our+information+which+can+allow+you+to+to+understand+paring+knife+earlier+than+you+truly+invest+cash+and+buy+some+knives.+We%27ve+got+specifically+targeted+%E2%80%98Paring+Knife%27+as+a+result+of+it+is+likely+one+of+the+mostly+used+knives+in+the+each+kitchen+across+the+globe+after+the+chef%27s+knife.%0D%0A%0D%0A%0D%0A%0D%0AThe+handle+is+another+high+point+of+the+device.+The+Fastback+II+has+a+very+deep+finger+groove+that+enables+for+an+especially+secure+grip.+This+is+helpful+when+it%27s+a+must+to+bear+down+on+the+knife%2C+like+if+you%27re+slicing+a+thick+cardboard+field+or+scoring+a+chunk+of+%5Bhttp%3A%2F%2FSheetrock.net%2F+sheetrock%5D+to+patch+a+gap+within+the+wall.+Simply+by+evenly+pinching+the+instrument+with+your+thumb+and+forefinger+along+with+your+forefinger+within+the+groove%2C+you+make+it+nearly+unimaginable+to+drag+the+knife+out+of+the+hand.%0D%0A%0D%0AThe+lengthy+serrated+knife+is+usually+referred+to+as+a+bread+knife+as+a+result+of+it+is+required+for+slicing+breads+that+require+a+backwards+and+forwards+sawing+motion.+Beyond+breads%2C+serrated+knives+are+good+for+tomatoes+and+peaches+and+different+fruits+which+have+a+skin+that+may+bruise+simply.++If+you+have+any+type+of+inquiries+relating+to+where+and+the+best+ways+to+utilize+%5Bhttp%3A%2F%2FWww.Thebestpocketknifereviews.com%2Fbest-utility-knife-top-rated-knives-analysis%2F+derma-safe+Folding+utility+knife%5D%2C+you+can+call+us+at+our+own+website.+However%2C+if+straight+edged+knives+%28such+because+the+paring+knife%29+are+saved+very+sharp%2C+they+may+be+able+to+do+slice+tomatoes+and+other+fruits+more+effectively+than+a+serrated+knife.+Though+many+people+assume+that+serrated+knives+cannot+be+sharpened%2C+this+is+not+true.+A+superb+sharpening+service+will+be+capable+to+sharpen+serrated+knives+with+out+problem.+Different+Knife+Types%0D%0A%0D%0ABut+before+we+depart+you+to+choose+your+individual+woodworking+utility+knife%2C+let%27s+just+caution+you+%E2%80%93+all+the+time+have+a+safety+glass+on+and+all+the+time+put+security+first.+With+that+mentioned%2C+now+you%27ll+be+able+to+head+over+to+Amazon%2C+and+choose+the+knives+that+greatest+fit+your+purpose+and+finances.+More+than+30+years+in+the+past+ZWILLING+J.A.+HENCKELS+revolutionized+the+standards+for+high-high+quality+knives+with+the+introduction+of+this+seamless+synthetic+handle.+This+masterpiece+of+security%2C+ergonomics+and+comfort+has+become+the+perfect+selling+ZWILLING+J.A.+HENCKELS+knife+series+worldwide.+Virtually+all+type+of+paring+knife+is+nice+and+a+%E2%80%98should+have+accent%27+in+kitchen+but+to+make+it+simpler+we+are+picked+up+few+that+we+just+have+to+point+out+as+our+favorite+knives+to+use.+Knife+Deal+with+Supplies+Oyster+Knife%0D%0A%0D%0ADiscovering+the+correct+knife+damascus+critiques+is+very+important+when+adorning+your+own+home.+However+you+do+not+need+to+spend+too+much%2C+so+it+is+also+essential+to+be+a+clever+buyer.+DHgate%27s+Shopping+for+Guides+provide+some+tips+that+will+help+you+select+the+exact+army+knifes+opinions+you+need.+I%27ve+every+top+rated+most+talked+about+in+style+survival+knives+made+%2C+this+device+stands+alone+not+even+to+be+spoken+about+with+the+others+holds+it+blade+like+no+othe+and+shall+be+giving+it+to+my+son+Cash+when+he%27s+of+age+Absolutely+wonderful%2C+this+knife+is+a+fcking+BEAST.+I+love+this+knife+so+much%2C+so+happy+I+picked+one+up%2C+bought+one+with+the+nylon%2Fcordura+and+it+is+just+superior.+I+can%27t+even+begin+to+explain+how+great+of+a+knife+this+is%2C+and+it+just+looks+beautiful.+Solid+10%2F10%0D%0A%0D%0ATo+get+the+full+range+of+use+out+of+the+knife+and+as+a+lot+security+and+portability+as+potential%2C+we+advocate+going+with+the+folding+type+over+the+retractable.+They%27re+smaller+to+store%2C+tend+to+return+with+belt+hooks%2C+and+because+of+the+way+the+folded+blade+nests+in+the+body%2C+there+is+a+decrease+probability+of+their+unintentionally+deploying+in+your+pocket.+Retractable+knives+are+good%2C+but+their+feature+set+is+extra+geared+toward+the+professional+tradesman.+Some+retractable+knives%2C+like+this+Snap-on+%2C+have+a+security+switch+that+needs+to+be+flipped+before+placing+the+knife+in+your+pocket%2C+however+why+bother+dealing+with+a+two-step+course+of+when+the+double-locking+folding+type+ensures+that+there+will+not+be+any+in-pocket+mishaps+simply+by+closing+the+knife%3F%0D%0A%0D%0AWhether+or+not+you+are+slicing+open+a+supply+from+Zappos+or+scoring+drywall+to+patch+a+hole+in+the+hallway%2C+the+Cala+Utility+Knife+handles+all+of+it.+Absolutely+retractable+for+control+and+security%2C+the+blade+glides+out+when+you%27re+working+and+hides+inside+while+you%27re+completed.+An+built-in+twine+cutter+makes+fast+work+of+brown+paper+packages+tied+up+with+string.+And+when+the+blade+will+get+boring%2C+you+have+obtained+five+razor-sharp+spares+stored+inside+acompartment.%0D%0A%0D%0AMake+sure+you+always+use+a+sharp+blade+when+chopping+leather%2C+hold+a+provide+of+Replacement+Blades+%233061-00+readily+available.+Use+%233461+Protecto+Board+or+%233464+Poly+Cutting+Board+as+a+base+to+chop+into.+This+will+help+shield+your+tools+and+your+desk+prime.+Woodcarving+is+a+time+tested+hobby+that+is+both+stress-free+and+fun.+Beginning+out+can+be+intimidating%2C+since+woodcarving+kits+come+with+so+many+little+instruments.+You+possibly+can+be+taught+the+talent+without+shopping+for+an+entire+set%3B+all+you+really+want+to+get+began+is+a+utility+knife+and+some+other+supplies.+After+making+an+attempt+your+hand+at+a+easy+carving%2C+you+can+think+about+pursuing+the+passion+further.+The+important+thing+to+having+good+knife+abilities+is+a+knife+you%27ll+be+able+to+maintain+withrelaxed+and+comfortable+grip+for+an+extended+period+of+time.+Reducing+Edge&create=Create dropped] on the floor and you cannot use them to pry with.<br><br>Ontario Knife Firm not solely makes military, sports activities and out of doors knives , however it is equally well-known for its comprehensive range of cutlery and instruments. It includes Old Hickory Kitchen Cutlery, Industrial and Agricultural Instruments, and Scientific and Medical Instruments and Devices. In addition to its in-house knife designs, Ontario has also tied up with famend knife designers akin [http://Www.thebestpocketknifereviews.com/best-utility-knife-top-rated-knives-analysis/ best Kitchen utility knife] to Bram Frank and Invoice Bagwell. The corporate gives many cheaper versions of Bagwell's exclusive designs. The again of the Pro Sheath (left) subsequent to the original Final Sheath (right). The Pro Sheath does away with both the Emergency Alerts information and the additional loops that permit for sideways carry.<br><br>Kendall would match right in at Rosewood Excessive. She's fairly, has great vogue sense and doubtless has a secret or two up her sleeve. As " Pretty Little Liars " fans know, Jackie is out to make Aria's life depressing, but the one place she can't get her is at school - that is, till her little sister (ender Kendall) enrolls at Rosewood High. Not solely does Kendall look like Jackie, but she may completely play the bitchy little sister. In spite of everything, we have seen her naughty facet on " Preserving Up With the Kardashians " various instances.<br><br>I know I should not should get into this but choose a knife that matches the scale of your hand! I do know somebody who carries round one of those obscenely long Spydercos! You must be capable of wrap your fist around the blade without it poking out. Don't decide up one thing that does not fit your frame, you won't carry it, and when you do, you get into aggressive look points (verify the law part beneath). Knives are quick for a cause, their power isn't within the how hard you can swing the blade like a sword, it is how solidly you can [http://ielaw.uibe.edu.cn/wiki/index.php?title=Superknife_Sk2_Folding_Utility_Knife manipulate] it together with your arm and hand strength.<br><br>The blade is razor sharp and made of D2 software steel. D2 is a very prime quality steel that has excessive wear resistance. It’ll keep sharp endlessly and is only used on the best knives. The blade includes a small blood groove and a well positioned thumb stud. The drop level and AXIS lock makes the knife really feel very robust and uniformed. Given its heavy obligation function, the blade performs exceptionally nicely. The Delica4 uses a David Boye Dent lock and this lock is a very, superb lock. Even underneath an excellent agency grip, the lock nonetheless holds. You'll be able to trust this lock even beneath probably the most demanding tasks.<br><br>SOG is a reputable US knife maker who has been producing quality knives designed for navy, regulation enforcement, and outdoor use since 1986. The company’s first knife was a redesign of the SOG Knife , a knife used by special forces troopers in Vietnam. At this time, nearly all of their products are designed with heavy obligation use in mind and the Aegis isn't any exception. The bone comes from Zebu cattle shin bones (you may read extra about it on Case’s website ), and sports traditional jigging with dark molasses coloring. This knife can be available with different shade and materials choices, corresponding to darkish purple bone, ebony wooden, and G-10.
 
:<math>P(r_1, r_2, \ldots, r_N) = 0.\ </math>
 
Strictly the ''X''<sub>''i''</sub> here are "non-commuting indeterminates", and so "polynomial identity" is a slight [[abuse of notation|abuse of language]], since "polynomial" here stands for what is usually called a "non-commutative polynomial". The abbreviation '''PI-ring''' is common. More generally, the free algebra over any ring ''S'' may be used, and gives the concept of '''PI-algebra'''.
 
If the degree of the polynomial ''P'' is defined in the usual way, the polynomial ''P'' is called '''monic''' if at least one of its terms of highest degree has coefficient equal to 1.
 
Every commutative ring is a PI-ring, satisfying the polynomial identity ''XY'' - ''YX'' = 0. Therefore PI-rings are usually taken as ''close generalizations of commutative rings''. If the ring has [[Characteristic (algebra)|characteristic]] ''p'' different from zero then it satisfies the polynomial identity ''pX'' = 0. To exclude such examples, sometimes it is defined that PI-rings must satisfy a monic polynomial identity.<ref>J.C. McConnell, J.C. Robson, ''Noncommutative Noetherian Rings, Graduate studies in Mathematics, Vol 30''</ref>
 
==Examples==
* For example if ''R'' is a [[commutative ring]] it is a PI-ring: this is true with
::<math>P(X_1,X_2)=X_1X_2-X_2X_1=0~</math>
 
* A major role is played in the theory by the '''standard identity''' ''s''<sub>''N''</sub>, of length ''N'', which generalises the example given for commutative rings (''N'' = 2). It derives from the [[Leibniz formula for determinants]]
 
::<math>\det(A) = \sum_{\sigma \in S_N} \sgn(\sigma) \prod_{i = 1}^N a_{i,\sigma(i)}</math>
 
:by replacing each product in the summand by the product of the ''X''<sub>i</sub> in the order given by the permutation &sigma;. In other words each of the ''N''! orders is summed, and the coefficient is 1 or &minus;1 according to the [[signature of a permutation|signature]].
 
::<math>s_N(X_1,\ldots,X_N) = \sum_{\sigma \in S_N} \sgn(\sigma) X_{\sigma(1)}\dotsm X_{\sigma(N)}=0~</math>
:The ''k''&times;''k'' [[matrix ring]] over any commutative ring satisfies a standard identity; the [[Amitsur&ndash;Levitzki theorem]] states that it satisfies ''s''<sub>2''k''</sub>. The degree of this identity is optimal since the matrix ring can not satisfy any monic polynomial of degree less than&nbsp;2''k''.
 
* Given a field ''k'' of characteristic zero, take ''R'' to be the [[exterior algebra]] over a [[countably infinite]]-dimensional [[vector space]] with basis ''e''<sub>1</sub>, ''e''<sub>2</sub>, ''e''<sub>3</sub>, ... Then ''R'' is generated by the elements of this basis and
 
::''e''<sub>''i''</sub>''e''<sub>''j''</sub> = &minus;''e''<sub>''j''</sub>''e''<sub>''i''</sub>.
 
:This ring does not satisfy ''s''<sub>''N''</sub> for any ''N'' and therefore can not be embedded in any matrix ring. In fact ''s''<sub>''N''</sub>(''e''<sub>1</sub>,''e''<sub>2</sub>,...,''e''<sub>''N''</sub>) =&nbsp;''N''!''e''<sub>1</sub>''e''<sub>2</sub>...''e''<sub>''N''</sub>&nbsp;&ne;&nbsp;0. On the other hand it is a PI-ring since it satisfies [[''x'',&nbsp;''y''],&nbsp;''z'']&nbsp;:=&nbsp;''xyz''&nbsp;&minus;&nbsp;''xzz''&nbsp;&minus;&nbsp;''zxy''&nbsp;+&nbsp;''zyz''&nbsp;=&nbsp;0. It is enough to check this for monomials in the ''e'''s. Now, a monomial of even degree commutes with every element. Therefore if either ''x'' or ''y'' is a monomial of even degree [''x'',&nbsp;''y''] :=&nbsp;''xy''&nbsp;&minus;&nbsp;''yx''&nbsp;=&nbsp;0. If both are of odd degree then [''x'',&nbsp;''y'']&nbsp;=&nbsp;''xy''&nbsp;&minus;&nbsp;''yx''&nbsp;=&nbsp;2''xy'' has even degree and therefore commutes with ''z'', i.e. [[''x'',&nbsp;''y''],&nbsp;''z'']&nbsp;=&nbsp;0.
 
==Properties==
* Any [[subring]] or [[homomorphism|homomorphic image]] of a PI-ring is a PI-ring.
* A finite [[direct product]] of PI-rings is a PI-ring.
* A direct product of PI-rings, satisfying the same identity, is a PI-ring.
* It can always be assumed that the identity that the PI-ring satisfies is [[multilinear]].
* If a ring is finitely generated by ''n'' elements as a [[module (mathematics)|module]] over its [[Center (algebra)|center]] then it satisfies every alternating multilinear polynomial of degree larger than ''n''. In particular it satisfies ''s''<sub>''N''</sub> for ''N''&nbsp;>&nbsp;''n'' and therefore it is a PI-ring.
* If ''R'' and ''S'' are PI-rings then their [[tensor product]] over the integers, <math>R\otimes_\mathbb{Z}S</math>, is also a PI-ring.
* If ''R'' is a PI-ring, then so is the ring of ''n''&times;''n''-matrices with coefficients in ''R''.
 
==PI-rings as generalizations of commutative rings==
Among noncommutative rings, PI-rings satisfy the [[Köthe conjecture]]. [[Finitely generated algebra|Affine]] PI-algebras over a [[field (mathematics)|field]] satisfy the [[Kurosh conjecture]], the [[Nullstellensatz]] and the [[catenary ring|catenary property]] for [[prime ideals]].
 
If ''R'' is a PI-ring and ''K'' is a subring of its center such that ''R'' is [[integral extension|integal over]] ''K'' then the [[Going up and going down|going up and going down properties]] for prime ideals of ''R'' and ''K'' are satisfied. Also the ''lying over'' property (If ''p'' is a prime ideal of ''K'' then there is a prime ideal ''P'' of ''R'' such that <math>p</math> is minimal over <math>P\cap K</math>) and the ''incomparability'' property (If ''P'' and ''Q'' are prime ideals of ''R'' and <math>P\subset Q</math> then <math>P\cap K\subset Q\cap K</math>) are satisfied.
<!--Add here results about integral extensions similar to commutative case.-->
 
==The set of identities a PI-ring satisfies==
 
If ''F''&nbsp;:=&nbsp;Z<''X''<sub>1</sub>, ''X''<sub>2</sub>, ..., ''X''<sub>''N''</sub>> is the free algebra in ''N'' variables and ''R'' is a PI-ring satisfying the polynomial ''P'' in ''N'' variables, then ''P'' is in the [[Kernel (algebra)|kernel]] of any homomorphism
:<math>\tau</math>:F <math>\rightarrow</math> ''R''.
 
An ideal ''I'' of ''F'' is called '''T-ideal''' if <math>f(I)\subset I</math> for every [[endomorphism]] ''f'' of ''F''.
 
Given a PI-ring, ''R'', the set of all polynomial identities it satisfies is an [[Ideal (ring theory)|ideal]] but even more it is a T-ideal. Conversely, if ''I'' is a T-ideal of ''F'' then ''F''/''I'' is a PI-ring satisfying all identities in ''I''. It is assumed that ''I'' contains monic polynomials when PI-rings are required to satisfy monic polynomial identities.
 
==References==
{{Reflist}}
*{{eom|id=P/p072640|title=PI-algebra|first=V.N.|last= Latyshev}}
*{{eom|id=a/a110570|title=Amitsur–Levitzki theorem |first=E.|last= Formanek}}
*[http://books.google.com/books?id=Li147JZ4T6AC Polynomial identities in ring theory], Louis Halle Rowen, Academic Press, 1980, ISBN 978-0-12-599850-5
*[http://books.google.com/books?id=x8gJw5bMw2oC Polynomial identity rings], Vesselin S. Drensky, Edward Formanek, Birkhäuser, 2004, ISBN 978-3-7643-7126-5
*[http://books.google.com/books?id=ZLW_Kz_zOP8C Polynomial identities and asymptotic methods], A. Giambruno, Mikhail Zaicev, AMS Bookstore, 2005, ISBN 978-0-8218-3829-7
*[http://books.google.com/books?id=80pw1QoLSQUC Computational aspects of polynomial identities], Alexei Kanel-Belov, Louis Halle Rowen, A K Peters Ltd., 2005, ISBN 978-1-56881-163-5
 
==External links==
*{{PlanetMath|urlname=PolynomialIdentityAlgebra|title=Polynomial identity algebra}}
*{{PlanetMath|urlname=StandardIdentity|title=Standard Identity}}
*{{PlanetMath|urlname=TIdeal|title=T-ideal}}
 
{{DEFAULTSORT:Polynomial Identity Ring}}
[[Category:Ring theory]]
[[Category:Mathematical identities]]

Latest revision as of 19:24, 1 January 2015



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