torsion unit group ring have been studi extens sinc 1960s associ scheme general group natur ask about torsion unit associ scheme ring paper establish some result about torsion unit associ scheme ring analog basic result torsion unit group ring

Allen Herman

Gurmail Singh

On the Torsion Units of Integral Adjacency Algebras of Finite Association Schemes

let ring eisenstein integ modulo paper studi zero divisor graph find diamet girth zero divisor graph character which graph complet complet bipartit bipartit regular eulerian hamiltonian chordal

Osama Alkam

Emad Abu Osba

Zero Divisor Graph for the Ring of Eisenstein Integers Modulo <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.2811508pt" id="M1" height="8.47856pt" version="1.1" viewBox="-0.0657574 -8.19741 9.14273 8.47856" width="9.14273pt"><g transform="matrix(.018,0,0,-0.018,0,0)"><path id="g113-111" d="M495 86L479 114C446 82 419 66 409 66C401 66 401 72 406 97C420 166 436 231 453 297C489 435 454 448 428 448C406 448 384 439 354 422C305 394 222 327 161 247H159L183 345C200 415 194 448 173 448C143 448 82 410 23 351L38 325C64 349 95 371 105 371C111 371 116 365 109 336L25 -4L31 -12C50 -4 77 3 107 9C119 69 132 122 145 168C197 254 321 381 370 381C387 381 393 374 378 305L329 95C309 17 320 -12 345 -12C372 -12 430 19 495 86Z"/></g></svg>

the concept idempot semir right weak regular semir right pure ideal semir introduc sever character them furnish

R. D. Jagatap

Right <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.2811995pt" id="M1" height="13.056pt" version="1.1" viewBox="-0.0657574 -12.7748 9.16067 13.056" width="9.16067pt"><g transform="matrix(.018,0,0,-0.018,0,0)"><path id="g113-108" d="M480 416C480 431 465 448 438 448C388 448 312 383 252 330C217 299 188 273 155 237H153L257 680C262 700 263 712 253 712C240 712 183 684 97 674L92 648L126 647C166 646 172 645 163 606L23 -6L29 -12C51 -5 77 2 107 8C115 62 130 128 142 180C153 193 179 220 204 241C231 170 259 106 288 54C317 0 336 -12 358 -12C381 -12 423 2 477 80L460 100C434 74 408 54 398 54C385 54 374 65 351 107C326 154 282 241 263 299C296 332 351 377 403 377C424 377 436 372 445 368C449 366 456 368 462 375C472 386 480 402 480 416Z"/></g></svg>-Weakly Regular <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.06579971pt" id="M2" height="11.9431pt" version="1.1" viewBox="-0.0657574 -11.8773 9.23249 11.9431" width="9.23249pt"><g transform="matrix(.018,0,0,-0.018,0,0)"><path id="g113-132" d="M485 501C480 550 475 615 475 650H42V622C119 616 127 606 127 525V126C127 44 119 34 41 28V0H311V28C220 34 212 43 212 126V584C212 612 216 615 244 615H318C382 615 404 608 420 587C435 568 445 541 455 497L485 501Z"/></g></svg>-Semirings

let topolog space the semigroup all étale map local homeomorph denot right left composit oper defin when composit oper inject the problem origin new approach studi étale polynomi map particular twodimension jacobian conjectur this approach construct fractal structur semigroup normal keller map outlin new method possibl attack open problem prepar the construct use left composit oper inject problem essenti paper complet solv inject problem two composit oper normal keller map also solv much easier surject problem composit oper

Ronen Peretz

Injectivity of the Composition Operators of &#xC9;tale Mappings

let commut field characterist let where two finit cyclic group give some structur result finit generat modul case where order divis extension modul also investig base extens same previous case show modul satisfi some condit have fair simpl form

Ahmed Najim

Mohammed Elhassani Charkani

Finitely Generated Modules over Group Rings of a Direct Product of Two Cyclic Groups

give iter close form formula hilbertkunz function ani binomi hypersurfac general over ani field arbitrari posit characterist prove hilbertkunz multipl associ ani binomi hypersurfac over ani field arbitrari posit characterist ration exampl also prove well known fact 1dimension binomi hypersurfac hilbertkunz multipl posit integ give precis account integ

Shyamashree Upadhyay

The Hilbert-Kunz Function for Binomial Hypersurfaces

obtain some elementari residu properti lattic modul obtain relat between weak primari element lattic modul weak prime element multipl lattic

C. S. Manjarekar

U. N. Kandale

Residuation Properties and Weakly Primary Elements in Lattice Modules

let ring say famili map jordan higher deriv map without assumpt addit ident map hold all each paper show everi jordan higher deriv map ring under certain assumpt becom higher deriv it applic get everi jordan higher deriv map banach algebra addit higher deriv

Mohammad Ashraf

Nazia Parveen

Jordan Higher Derivable Mappings on Rings

the set all polynomi over multipl hyper form commut group respect componentwis addit polynomi for polynomi set polynomi whose compon chosen from set where compon respect multipl hyper polynomi structur hyperstructur multipl ring the purpos paper studi properti multipl ring correspond those polynomi structur multipl hyper

Utpal Dasgupta

Some Properties of Multiplicative <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-4.5247pt" id="M1" height="16.402pt" version="1.1" viewBox="-0.0657574 -11.8773 20.3678 16.402" width="20.3678pt"><g transform="matrix(.018,0,0,-0.018,0,0)"><path id="g113-73" d="M828 650H570L564 622C646 614 650 609 634 524L604 366H253L281 524C296 608 308 615 382 622L388 650H132L126 622C211 615 214 609 198 524L121 122C106 42 102 35 23 28L17 0H276L282 28C196 35 191 40 206 122L243 324H597L559 123C544 42 537 35 452 28L446 0H710L716 28C632 35 628 43 643 122L719 524C735 610 741 615 822 622L828 650Z"/></g><g transform="matrix(.013,0,0,-0.013,13.86,4.308)"><path id="g185-40" d="M219 86C216 168 211 250 206 337C201 410 189 448 163 448C131 448 79 396 43 344L60 322C91 359 110 375 118 375S132 358 136 298C141 238 152 81 155 -12H182C242 62 331 177 390 258C435 321 451 360 451 391C450 424 432 448 408 448C390 448 372 435 366 419C362 410 362 401 366 394C373 383 376 367 376 350C376 283 262 138 221 86H219Z"/></g></svg>-Rings of Polynomials over Multiplicative Hyperrings

investig prime primari element compact generat multipl lattic counterexampl shown primari element need primari some character primari prime element obtain final some result almost prime almost primari element character obtain

A. V. Bingi

<svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-4.4457pt" id="M1" height="16.7179pt" version="1.1" viewBox="-0.0657574 -12.2722 10.6865 16.7179" width="10.6865pt"><g transform="matrix(.018,0,0,-0.018,0,0)"><path id="g5-243" d="M567 249C567 371 489 431 403 469C398 471 396 479 400 499L436 676L417 685L355 650L235 60C197 75 150 137 150 221C150 338 202 389 277 428L260 462C128 416 21 344 21 204C21 90 98 9 220 -10L200 -117C183 -207 215 -239 244 -246L250 -243L298 -8C425 2 567 94 567 249ZM478 235C478 131 407 46 313 47L382 397C413 387 478 326 478 235Z"/></g></svg>-Prime and <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-4.4457pt" id="M2" height="16.7179pt" version="1.1" viewBox="-0.0657574 -12.2722 10.6865 16.7179" width="10.6865pt"><g transform="matrix(.018,0,0,-0.018,0,0)"><use xlink:href="#g5-243"/></g></svg>-Primary Elements in Multiplicative Lattices

prove free group relat varieti all group simultan nilpot class most metabelian central everi noncentr element abelian relat result model theori group well quest find relat analog classic theorem benjamin baumslag also touch briefli similar consider varieti nilpot group

Anthony M. Gaglione

Seymour Lipschutz

Dennis Spellman

The Relatively Free Groups <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-4.524599pt" id="M1" height="20.4677pt" version="1.1" viewBox="-0.0657574 -15.9431 81.0923 20.4677" width="81.0923pt"><g transform="matrix(.018,0,0,-0.018,0,0)"><path id="g113-71" d="M584 650H137L131 622C214 614 217 612 200 521L125 127C109 41 101 35 23 28L17 0H288L294 28C201 35 193 42 209 128L242 309H348C440 309 442 300 443 226H471L510 422H482C452 354 449 348 357 348H251L295 575C302 609 304 615 338 615H426C502 615 517 604 526 581C534 560 536 524 537 492L565 494C574 554 583 631 584 650Z"/></g><g transform="matrix(.018,0,0,-0.018,10.861,0)"><path id="g113-41" d="M300 -147C201 -63 143 98 143 270S200 602 300 686L282 710C136 610 70 450 70 271V270C70 89 136 -72 282 -170L300 -147Z"/></g><g transform="matrix(.018,0,0,-0.018,17.049,0)"><path id="g113-79" d="M822 650H589L583 622C660 617 677 607 674 561C672 534 664 481 647 390L600 137H596L273 650H126L120 622C176 620 194 615 207 594C221 571 225 557 214 504L161 257C141 166 129 112 121 85C108 42 83 30 29 28L23 0H260L266 28C193 33 173 42 176 89C178 122 186 172 202 255L256 527H259L583 -8H612L690 390C708 481 720 535 728 558C744 603 756 619 816 622L822 650Z"/></g><g transform="matrix(.013,0,0,-0.013,30.891,4.308)"><path id="g50-100" d="M387 400C387 425 348 451 303 451C247 451 176 414 132 376C69 322 24 228 24 148C24 43 74 -12 147 -12C211 -12 301 33 363 103L346 128C319 99 249 51 193 51C148 51 112 84 112 165C112 230 130 287 154 330C170 359 199 400 243 400C277 400 304 383 326 354C333 345 343 343 354 348C378 360 387 382 387 400Z"/></g><g transform="matrix(.018,0,0,-0.018,40.71,0)"><path id="g117-63" d="M574 0L325 548H263L14 0H76L293 479H295L512 0H574Z"/></g><g transform="matrix(.018,0,0,-0.018,55.204,0)"><path id="g113-66" d="M686 28C612 35 607 44 591 112C563 234 541 360 519 489L489 666L457 658L147 121C100 40 89 36 24 28L17 0H240L250 28C168 34 159 41 190 101L262 237H482C495 180 503 137 510 91C517 47 514 35 441 28L433 0H677L686 28ZM475 280H285L429 541H431L475 280Z"/></g><g transform="matrix(.013,0,0,-0.013,67.77,-7.899)"><path id="g50-51" d="M414 144C384 79 371 75 317 75H135L276 221C367 316 408 376 408 465C408 570 327 635 237 635C179 635 131 609 100 575L42 494L67 471C94 510 138 565 205 565C277 565 321 517 321 435C321 348 258 270 195 195C146 137 88 81 33 26V0H411C423 44 433 88 446 135L414 144Z"/></g><g transform="matrix(.018,0,0,-0.018,74.548,0)"><path id="g113-42" d="M275 270C275 450 212 609 64 710L45 686C145 604 203 442 203 270S147 -63 45 -147L64 -170C213 -68 275 89 275 270Z"/></g></svg> Satisfy Noncentral Commutative Transitivity

let let algebra close field characterist 0 greater than show satisfi simultan triangulariz let reduc ring zero divisor let generic matrix over show sole solut let commut ring uniti let similar everi zero divisor nilpot solut where

Gerald Bourgeois

The Matrix Equation <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-4.7149pt" id="M1" height="18.4088pt" version="1.1" viewBox="-0.0657574 -13.6939 149.427 18.4088" width="149.427pt"><g transform="matrix(.018,0,0,-0.018,0,0)"><path id="g113-89" d="M748 650H522L515 622L546 617C580 611 587 604 565 575C518 513 469 451 419 393C376 474 349 534 330 580C318 609 325 612 361 618L383 622L392 650H151L144 622C214 616 224 612 257 543L360 327C270 218 187 124 159 95C106 40 92 34 26 28L17 0H252L259 28L236 31C189 37 188 47 209 78C249 136 308 210 377 294L478 79C494 44 487 37 449 32L418 28L409 0H673L680 28C596 34 591 39 554 116L436 361C526 469 574 521 604 553C659 612 669 614 739 622L748 650Z"/></g><g transform="matrix(.018,0,0,-0.018,13.143,0)"><path id="g113-66" d="M686 28C612 35 607 44 591 112C563 234 541 360 519 489L489 666L457 658L147 121C100 40 89 36 24 28L17 0H240L250 28C168 34 159 41 190 101L262 237H482C495 180 503 137 510 91C517 47 514 35 441 28L433 0H677L686 28ZM475 280H285L429 541H431L475 280Z"/></g><g transform="matrix(.018,0,0,-0.018,29.707,0)"><path id="g117-33" d="M535 230V280H52V230H535Z"/></g><g transform="matrix(.018,0,0,-0.018,44.202,0)"><use xlink:href="#g113-66"/></g><g transform="matrix(.018,0,0,-0.018,56.768,0)"><use xlink:href="#g113-89"/></g><g transform="matrix(.018,0,0,-0.018,75.445,0)"><path id="g117-34" d="M535 323V373H52V323H535ZM535 138V188H52V138H535Z"/></g><g transform="matrix(.018,0,0,-0.018,90.939,0)"><use xlink:href="#g113-89"/></g><g transform="matrix(.013,0,0,-0.013,104.621,-7.899)"><path id="g50-223" d="M558 106L533 129C501 84 476 67 464 67C446 67 435 116 414 239C457 297 508 367 550 440L539 451L484 438C458 390 429 338 404 300H402C375 407 351 451 285 451C168 451 24 312 24 153C24 56 66 -12 132 -12C206 -12 284 64 345 156H347C367 24 386 -12 422 -12C457 -12 503 7 558 106ZM338 206C272 99 215 56 173 56C140 56 116 96 116 170C116 303 194 405 255 405C304 405 324 299 338 206Z"/></g><g transform="matrix(.018,0,0,-0.018,112.591,0)"><path id="g113-104" d="M546 430L539 434C529 434 505 438 495 440C473 444 450 448 430 448C352 448 265 412 213 366C145 306 96 203 96 103C96 22 135 -12 160 -12C190 -12 238 14 262 32C310 68 368 120 411 184H413C403 117 396 75 384 21C353 -118 325 -158 291 -184C270 -200 241 -205 208 -205C133 -205 90 -164 74 -110C70 -98 58 -100 49 -107C34 -119 23 -140 23 -155C23 -190 74 -261 166 -261C219 -261 280 -233 314 -208C383 -157 446 -79 470 81C491 223 529 388 546 430ZM456 386C452 357 433 283 420 252C402 216 366 174 325 129C288 88 239 56 212 56C192 56 182 77 182 120C182 165 199 242 226 292C256 348 281 377 311 389C327 395 353 402 375 402C408 402 436 394 456 386Z"/></g><g transform="matrix(.018,0,0,-0.018,122.768,0)"><path id="g113-41" d="M300 -147C201 -63 143 98 143 270S200 602 300 686L282 710C136 610 70 450 70 271V270C70 89 136 -72 282 -170L300 -147Z"/></g><g transform="matrix(.018,0,0,-0.018,128.956,0)"><use xlink:href="#g113-89"/></g><g transform="matrix(.018,0,0,-0.018,142.637,0)"><path id="g113-42" d="M275 270C275 450 212 609 64 710L45 686C145 604 203 442 203 270S147 -63 45 -147L64 -170C213 -68 275 89 275 270Z"/></g></svg> over Fields or Rings

introduc concept almost semiprim submodul unitari modul over commut ring nonzero ident investig some basic properti almost semiprim weak semiprim submodul give some character them especi finit generat faith multipl modul

Farkhonde Farzalipour

On Almost Semiprime Submodules

swan theorem determin pariti number irreduc factor binari trinomi work studi pariti number irreduc factor special binari pentanomi even degre where exact one odd this kind irreduc pentanomi can use fast implement trace squar root comput finit field characterist 2

Ryul Kim

Yun Mi Kim

The Reducibility of a Special Binary Pentanomial

consid union parafre lie algebra prove union again parafre under some condit

Naime Ekici

Zehra Velioğlu

Unions of Parafree Lie Algebras

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Algebra