Involution on Rings

Available online at: http://www.ijmtst.com/vol5issue04.html International Journal for Modern Trends in Science and Technology ISSN: 2455-3778 :: Volume: 05, Issue No: 04, April 2019 Involution on rings P Sreenivasulu Reddy1 | Kassaw Benebere2 1,2 Department of Mathematics, Samara University, Semera, Afar Regional State, Ethiopia. To Cite this Article P Sreenivasulu Reddy and Kassaw Benebere, “Involution on rings”, International Journal for Modern Trends in Science and Technology, Vol. 05, Issue 04, April 2019, pp.-51-55. Article Info Received on 07-March-2019, Revised on 12-April-2019, Accepted on 19-April-2019. ABSTRACT This paper deals with the concept of involution rings, involution ring properties and some important results and we discussed some important results in simple rings, minimal and maximal ideals with involution. Copyright © 2019 International Journal for Modern Trends in Science and Technology All rights reserved. I. INTORDUCTION an involution of 푆 satisfying the identities ∗ ∗ ∗ ∗ ∗ ∗ 푥 + 푦 = 푥 + 푦 , 푥푦 = 푦 푥 Definition1.1.An involution on ring 푅is a unary Sometimes, semi ring may be equipped with a zero, operation * on 푎∗ ∗ = 푎 , 푎 + 푏 ∗ = 푎∗ + 푏∗ and and or an identities one. In such a case we require ∗ ∗ ∗ 푎푏 = 푏 푎 for each 푎, 푏 ∈ 푅 . A ring with this that the involution of satisfies 0∗ = 0 and 1∗ = 1. opration is called a ring with involution or *-ring. Definition1.6.A right (left) ideals of 퐼∗ in a ring 푅∗ Definition1.2.Let 푅 be a *-rings and let 퐼 be an is called *- minimal ideals if ∗ ideal of a ring 푅. Then 퐼 is a *-ideal if 퐼 is sub set of i. 퐼∗ ≠ 0 ∗ and 퐼.(here we define for any non empty set 푆, we define ii. If 퐽∗ is a non zero (right/left) ideal of a ∗ ∗ 푠 = {푠 : 푠 ∈ 푆}) ring 푅 with involution contained in 퐼∗ Definition1.3.A non zero ideal 퐼 of an involution then 퐼∗ = 퐽∗. ring 푅 which is closed under involution is termed as *-ideal.That is 퐼∗ = {푎∗ ∈ 푅|푎 ∈ 퐼} ⊆ 퐼 Definition1.7.For a *-ring R the direct sum of all Definition1.4.Asub ring 퐴 of 푅 is said to be a minimal *- ideal is denoted by 푆표푐∗(푅), the *- scole bi-ideal of 푅 if 퐴푅퐴 ⊆ 퐴 and a *-bi ideal if in of R. (if R has no *- minimal ideals then define addition it is closed under involution of 푅. 푆표푐∗(푅) to be zero).and also we have 푆표푐∗(푅) is the Example let 푆 be a ring and let 푋 be an ideal of 푆 sub set of 푆표푐(푅). 푆 푋 Definition1.8.Wesay R is *- simple R has no non with푋2 = 0. Let 푅 = then with the standard 푋 0 zero proper ideals. matrix operation 푅 is a ring and a matrix Definition1.9.An ideal 퐴∗ is said to be maximal transposition is an involution, denoted by *. If 푆 ideals of the ring R with involution *, if 2 has a unity, then푅 = 푅. If 푋 is a minimal ideal of i. 퐴∗ ≠ 푅∗ 0 푋 푆,then 푋 = is a minimal ideal of 푅. ii. For anyideals 퐵∗ is a sub set of 퐴∗ , 0 0 ∗ ∗ ∗ ∗ Definition1.5.(푆, +, . ) is a semi- ring (a structure in either 퐵 = 퐴 or 퐵 = 푅 which (푆, +) is a commutative semi group , 푆, . is Definition1.10.LetR be a ring with involution. For a semi group, and distributes over +) then (푆, +, . ) some 푎 ∈ 푅, one may write is called an involution semi ring provided that *is 51 International Journal for Modern Trends in Science and Technology P Sreenivasulu Reddy and Kassaw Benebere : Involution on rings 푖 퐼 =< 푎 >∗= 푍푎 + 푍푎∗ + 푎푅 + 푅푎 + 푅푎푅 + In part (ii) when 퐼2 ≠ 0 we have 퐾2 ≠ 0. Recall that a 푎∗푅 + 푅푎∗ + 푅푎∗푅. minimal ideal which is not square zero is a simple ii Clearly I is an ideal of R closed under ring, so in the case we have 퐾 and 퐾∗ are simple involution and is called the principal *- ideal rings. From 퐼 = 퐾 ⊕ 퐾∗ and 퐼\퐾∗ and 퐼\퐾∗ ≈ 퐾 we generated by 푎. One may deduce that 퐼 =< 푎 > see that 퐾∗, and hence 퐾are maximal ideals of the +< 푎∗ > = < 푎, 푎∗ >. ring 푅 . And in similar situation in the iii. A ring with involution * is a principal *- square,occur frequently enough to warrant looking ideal ring if each *- ideal is a principal *- more closely at the setting where the ring is the ideal. More over we say that a group G is direct sum of two simple rings. Ideal of 푅 and is strongly principal *- ideal ring group, if contained 퐼. So case(i) must hold when 푅2 = 0. G is not nil (퐺 ≠ 0) and every ring R Preposition1.15.Let푅 be a *- ring and let 퐼 be a with involution satisfaying 푅2 ≠ 0, and minimal ideal of 푅 such that 퐼2 ≠ 0. Then either 퐺 = 푅∗ is a principal *- ideal ring. (exclusively) i. 퐼 is *- ideal of 푅; or Definition1.11.Let R be *- ring. The intersection of ii. 퐼 ⊕ 퐼∗ is a *- simple ring and hence is a all the nonzero ideals of R, called heart of R. and is *- minimal ideal of 푅. denoted by H(R) so R is sub directly irreducible if and only if 퐻(푅) ≠ 0. The intersection of all nonzero Proof:Assumesthat 퐼 ≠ 퐼∗ . Then minimally of 퐼 *- ideals of R is denoted by 퐻∗(푅). Then R is *- sub yields 퐼 ∩ 퐼∗ = 0 and hence 퐼 + 퐼∗ = 퐼 ⊕ 퐼∗. Since퐼2 ≠ directly irreducible if and only if 퐻∗(푅) ≠ 0 . If 0, the minimal ideals 퐼 and 퐼∗ are simple rings and 퐻∗ 푅 = 0, then R is a sub direct product of *- sub the rings 퐼 ⊕ 퐼∗has only 퐼 and 퐼∗ as proper nonzero direct of irreducible ring. ideals. Let 푌 be a non zero proper *- ideal of the Definition1.12.A *- ideal is said to be *- prime if *-ring 퐼 ⊕ 퐼∗ . So 푌 is 퐼 or 퐼∗ since 푌 is invariant when ever 퐴, 퐵 are *- ideals such that 퐴퐵 ⊆ 푃 under *, either case leads to the contradictory퐼 = 퐼∗. then퐴 ⊆ 푃, 표푟 퐵 ⊆ 푃. From this R is *- prime ring if So 퐼 ⊕ 퐼∗ is a *- simple and consequently the only and only if 0 is a *- prime ideal. non zero *- ideal of 푅 contained 퐼 ⊕ 퐼∗ is itself. Corollary1.13.In everyinvolution ring Example:Let Sbe a ring and let X be an ideal of S 2 2 푅, 푛푅, 푅 푛 , 푅푡 , 푅푝 and the maximal divisisble ideal 퐷 with 푋 = 0. If S has unity, then 푅 = 푅. If X is 0 푋 are *-ideals. minimal ideal of S, then 푋 = is minimal Proposition1.14.Let 푅 be a *- ring. if 퐼 is 0 0 ideal of R. observe that 푋 = 0, 푋 ⊕ 푋∗is a *- minimal *-minimal ideals of 푅 ,then either ideal of R. i. 퐼 is a minimal ideal of 푅 ; or Preposition1.16.Let R be a *- ring. Then푆표푐 푅 = ii. For any non zero ideal 퐾 of 푅, with 퐾 푇 ⊕ 푆표푐∗ 푅 = 푇∗ ⊕ 푆표푐∗(푅), where T is the direct proper sub set of I,then 퐼 = 퐾 ⊕ sum of minimal ideals K of R such that K2 = 0 and 퐾∗,퐾&퐾∗ are minimal ideals of 푅.further 퐾 ⊕ 퐾∗ is not a *- minimal ideal of R. more 퐼2 ≠ 0,then there are exactly two Proof:Using a standard argument it can be shown non zero ideals of 푅 properly contained that 푆표푐(푅) is a direct sum of minimal ideals. Label in 퐼. If 푅2 = 0, then only case(i) this summands with ordinals so that 푆표푐 푅 = ∗ Proof:Assume that 퐼 is not minimal ideal of 푅 and ⊕ 푀ƛ , ƛ ∈ 퐴 and 푆표푐 (푅) ∗ let 퐾 be any non zero ideal of 푅 with 퐾 ⊆ 퐼 since 훼≤ƛ ⊕ 푀ƛ , 푇 = 훼≤ƛ ⊕ 푀ƛ ,. Then 푆표푐 푅 = 푆표푐 (푅) ⊕ ∗ ∗ 2 퐾 + 퐾 and 퐾 ∩ 퐾 are *- ideals of R and contained 푇. If 퐼 is minimal ideal of Rand I ≠ 0 then I is the ∗ ∗ ∗ ∗2 in 퐼 .we have 퐾 + 퐾 = 퐼 and 퐾 ∩ 퐾 = 0 . So sub set of 푆표푐 (푅). Then 훼 ≤ ƛ we have 푀ƛ = 0 and ∗ ∗ 퐼 = 퐾 ⊕ 퐾 .let 푌 be non zero ideal of 푅 such that 푌 푀ƛ + 푀ƛ is not a *-minimal ideals of R. since ∗ ∗ ∗ a sub set of 퐾 then by the above argument,퐼 = 푌 ⊕ T ∩ Soc R = 0 we have T ∩ Soc R = 0 and ∗ ∗ ∗ 푌 for any 푘 ∈ 퐾 we have 푘 = 푦1 + 푦2 where consequently푆표푐 푅 = Soc R ⊕ 푇 . ∗ 푦1 ∈ 푌, 푦2 ∈ 푌 and hence 푘 − 푦1 = 푦2 thus 푦2 ∈ 퐾 ∩ Proposition1.17.Let R be *- simple ring. Then 퐾∗ = 0,yielding 푘 ∈ 푌 and hence 퐾 = 푌 so 퐾&퐾∗ are either R is simple or R contains a maximal ideal K minimal ideals of 푅. Next let 퐵 be non zero ideal of such that 푅 = 퐾 ⊕ 퐾∗ , 퐾 and 퐾∗ are simple rings 푅 such that 퐵 ⊆ 퐼 and 퐵neither 퐾 nor 퐾∗ .as with 푅2 ≠ 0, and then the only proper non zero ideals of the argument for 퐾, 퐵 is minimal ideal of 푅 . So R are 퐾 and 퐾∗. 퐵퐾 = 0 . Similarly, 퐵퐾∗, 퐵∗퐾 and 퐵∗퐾∗ are zero, Proof:Observe that if R is not simple, then R is *- yielding 퐼2 = 퐵 ⊕ 퐵∗ 퐾 ⊕ 퐾∗ = 0 when 푅2 = 0 the minimal ideal of itself which is not a minimal ideal. set {푘 + 푘∗: 푘 ∈ 퐾} is a *- ideal of 푅.

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