Involution on Rings

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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.

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) ⊕ 퐾 + 퐾∗ and 퐾 ∩ 퐾∗ are *- ideals of R and contained 푇. If 퐼 is minimal ideal of Rand I2 ≠ 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. So 푅 = 퐾 ⊕ 퐾∗,where K and 퐾∗ are minimal ideals

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P Sreenivasulu Reddy and Kassaw Benebere : Involution on rings

of R. if T is an ideal of the ring K then T is an ideal Theorem1.19.Let R be *- ring. If M is a maximal of , so minimality of K yields 푇 = 0 or 푇 = 퐾. Thus K ideal of the ring R, then either (exclusively); is a simple ring. Since 푅/푅∗ ≈ 퐾, we have 퐾∗is a i. 푀 ∩ 푀∗ is a maximal ideal of R; or maximal ideal of R. similarly we obtain 퐾∗ is a ii. For any *- ideal K of R such that simple ring and K is a maximal ideal of R. since 푀 ∩ 푀∗ ⊆ 퐾 ⊆ 푅 , then K is a maximal 푅2 = 0 implies the existence of non zero proper *- ideal of R and there is no ideal of R ideal of R, we must have 푅2 ≠ 0 then K and 퐾∗ the properly contained between 푀 ∩ 푀∗ and only non zero proper ideals of R. K, and between 푀 ∩ 푀∗ and M. Theorem1.18.Let R be *- ring and let M be a *- ∗ maximal ideal of R. if M is not maximal ideal of R, Proof:(i). Assume that 푀 ∩ 푀 is not maximal ∗ then there exist a maximal ideal K of R such that; ideal of R. then 푀 ≠ 푀 . let K be any *-ideal of R ∗ ∗ i. 퐾 + 퐾∗ = 푅 and 퐾 ∩ 퐾∗ = 푀; so 푅\퐾∗ is a such that 푀 ∩ 푀 ⊆ 퐾 ⊆ 푅 . From 푅\푀 = (푀 + ∗ ∗ ∗ ∗ simple and 푅\푅∗ ≈ 퐾\푀 푀 )\푀 ≈ 푀(푀 ∩ 푀 ), we have that 푀(푀 ∩ 푀 ) is a ∗ ii. 푅2 + 푀 = 푅 and the only proper ideal of simple ring. Thus 푀 ∩ 푀 is a maximal ideal of the R which properly contains M are K and ring M and there are no ideals of R properly ∗ ∗ 퐾∗; contained between 푀 ∩ 푀 and M. from 푀 ∩ 푀 sub iii. K and 퐾∗ are *- essential in R; set of 푀 ∩ 퐾 also sub set of M, we have either ∗ iv. Either M is essential as an ideal in the 푀 ∩ 퐾 = 푀 or 푀 ∩ 퐾 = 푀 ∩ 푀 . The former yields M ring R, or ther exist a maximal ideal of R sub set of K and hence 푀 = 퐾, a contradiction to M ∗ such that 퐾∗ = 푀 ⊕ 퐼 and 푅 = 퐼 ⊕ 퐼∗ ⊕ is *- ideal. Consider 푀 ∩ 퐾 = 푀 ∩ 푀 .then K is not ∗ 푀. sub set of M and 푅\퐾 = (푀 + 퐾)\퐾 ≈ 푀(푀 ∩ 푀 ) . Thus 푅\퐾 is a simple ring and consequently K is Proof:Since 푅 = 푅\푀 is a *- simple ring which is maximal ideal of R. from 푅\푀 = (퐾 + 푀)\푀 ≈ not simple, we have that there exists a proper ideal 퐾\(퐾 ∩ 푀) = 퐾(푀 ∩ 푀∗), we see that 퐾\(푀 ∩ 푀∗) is a K of R with M proper sub set of K such that simple ring and hence there are no ideals of R 퐾\푀 ⊕ 퐾\푀 ∗, with 퐾\푀 and 퐾\푀 ∗ = 퐾∗\푀 being properly contained in between 푀 ∩ 푀∗ and K. both maximal as ideals are and simple as rings. Lemma1.20.Let 퐺 = 퐻 ⊕ 퐾 , 퐻 ≠ 0, 퐾 ≠ 0 be a Thus K and 퐾∗ are maximal ideals of R, 퐾 + 퐾∗ = 푅, strongly principal *- ideal ring group. Then H and K 퐾 ∩ 퐾∗ = 푀 since 푅2 = 푅, the only proper non zero are either both *- cyclic or both nil. ideals of R are 퐾\푀 and 퐾∗\푀. Then there are no Proof:Suppose that H is not nil. Let S be a *- ring proper ideals of R that properly contain 푀 except K with 푆+ = 퐻 and 푆2 ≠ 0 and let T be the zero ring on and 퐾∗ .For a purpose of contradiction assume K. the ring direct sum 푅 = 푆 ⊕ 푇 is the ring with there exist a non zero *-ideal T of R such that involution satisfaying 푅+ = 퐺 and 푅2 ≠ 0. Since T is 푇 ∩ 퐾 = 0 also 푇 ∩ 퐾∗ = 0 . Then 푇퐾 = 0 = 푇퐾∗ and a *- ideal in R, 푇 =< 푥 >∗ . Clearly 퐾 = 푇+ = 푥 ∗ . hence 푇푅 = 푇 퐾 + 퐾∗ = 푇퐾 + 푇퐾∗ = 0 . Since Therefore K is not nil interchanging the roles of H 푇 ∩ 퐾 = 0 also we have 푅 = 푀 + 푇 = 푀 ⊕ 푇. Hence and K yields that His *- cyclic. 푅\푀 ≈ 푇, and 푇 is a *- simple ring which not a Corollary1.20.Let 푮 = 퐻 ⊕ 퐾, 퐻 ≠ 0, 퐾 ≠ 0 be simple ring.also we have푇2 ≠ 0, a contradiction to strongly principal *- ideal ring group. Then H and K 푇푅 = 0 . Thus 퐾∗ is *- essential in R, which are *- cyclic. immediately yields that 퐾∗ also.To establish (iv) Proof:It suffices to negate that H and K are both consider M to be not essential in the set of ideals of nil. Let 푅 = 퐺 . be a ring with involution the ring R. observe that if K is essential in the set of satisfying푅2 ≠ 0. ∗ 2 ideals of the ring R, then so 퐾 , and consequently Case (i): suppose that 푅 ⊆ 퐾 . There exist 푕0 ∈ ∗ ∗ the non zero intersection 푀 = 퐾 ∩ 퐾 would also 퐻, 푘0 ∈ 퐾, such that 푅 =< 푕0, 푘0 > . Let 푕 ∈ 퐻, since be essential. Thus there exists a non zero ideal I of 푕 ∈ 푅 there exist integers 푛 and 푚, and 푥 ∈ 푅2 such R such that 퐼 ∩ 퐾 = 0. The only ideals of R which that ∗ ∗ ∗ contains M are M,K,퐾 and R. so either 퐼 ⊕ 푀 = 퐾 푕 = 푛 푕0 + 푘0 + 푚 푕0 + 푘0 + 푥However , 푥 ∈ 퐾, so ∗ ∗ or 퐼 ⊕ 푀 = 푅. In the former case, from 퐼 ≈ 푀\퐾 we 푕 = 푛푕0 + 푚푕0 and H is *-cyclic, contradicting the have that I is a simple ring and hence I is a minimal fact that H is nil. 2 ideal of R. finally consider 퐼 ⊕ 푀 = 푅 for each 푘 ∈ 퐾 Case (ii): suppose that 푅 ≠ 퐾 . For all 푔1, 푔2 ∈ 퐺 there exists ∈ 퐼 , 푚 ∈ 푀 such that 푘 = 푖 + 푚 . But define 푔1표푔2 = 휋퐻 (푔1표푔2) , where 휋퐻 is natural 푖 = 푘 − 푚 is in K, forcing 푖 = 0. So 푘 = 푚. Since 푘 is projection of G on to H. since ∗ ∗ ∗ ∗ ∗ arbitrarily this yields the contradictory 푀 = 퐾.Thus 푔1표푔2 = (휋퐻 푔1표푔2) = 휋퐻 푔1표푔2 = 휋퐻 푔2표푔1 ∗ ∗ 퐼 ⊕ 푀 ≠ 푅. = 푔2표푔1

53 International Journal for Modern Trends in Science and Technology

P Sreenivasulu Reddy and Kassaw Benebere : Involution on rings

Hence 푆 = 퐺, 표 is a ring with involution 2) Conversely, suppose that 퐺푡 is bounded, and satisfying 푆2 ⊆ 퐻 . The argument employed in (i) that there exists a unitals principals *- ideals ring yield that K is *- cyclic with contradicts the fact S with unity and * is the identity involution such + that K is nill. that 푆 = 퐺푡 . Let 푅 = 푆퐼 = 퐼 ∩ 푆 ⊕ 퐼 ∩ 푇 . T with Theorem1.21.There are no mixed strongly 푒, 푓 the unities of 푆 and 푇, respectively. Then R is a principal *- ideal ring groups. ring with involution *. Proof:Let G be a mixed strongly principal *- ideal Let I be a *- ideal in R, then 퐼 = 퐼 ∩ 푆 ⊕ (퐼 ∩ 푇) ring group. G is decomposable, so by lamma (that Now 퐼 ∩ 푆힓∗푺 and so 퐼 ∩ 푆 =< 푥 >∗ . Similarly is let 퐺 = 퐻 ⊕ 퐾, 퐻 ≠ 0, 퐾 ≠ 0 be strongly principal 퐼 ∩ 푇 =< 푦 >∗. *- ideal ring group. Then H and K are either both *- Clearly < 푥 + 푦 >∗⊆ 퐼 however, 푥 = 푒 푥 + 푦 ∈< 푥 + cyclic or both nill. So 퐺 = 퐻 ⊕ 퐾, 퐻 ≠ 0, 퐾 ≠ 0 with H 푦 >∗, 푥∗ = 푒 푥 + 푦 ∗ ∈< 푥 + 푦 >∗, and 푦 = 푓 푥 + 푦 ∈< and K both *- cyclic or both nill. 푥 + 푦 >∗, 푦∗ = 푓 푥 + 푦 ∈< 푥 + 푦 >∗ 1) Suppose that H and K are both nill. There Hence we conclude that 퐼 =< 푥 + 푦 >∗. are no mixed nil groups by so, we may Theorem1.23.Let G be a torision free strongly *- assume that H is a torsion group, and that ideal ring group. Then G is either indecomposable, K is torision free. Let R be *- ring with or is the direct sum of two nil groups. + 2 ∗ 푅 = 퐺 and 푅 ≠ 0. Clearly H is a *- ideal in Proof:It suffices to negate that 퐺 = 푥1 ⊕ ∗ ∗ R and so 퐻 =< 푕 > . 푥2 , 푥푖 ≠ 0, 푖 = 1,2 Suppose that it is so the product : 푥푖 푥푗 = 3푥푖 and Let 푕 = 푚, then 푚퐻 = 0. H is divisible, there fore ∗ ∗ 푥푖 푥푗 = 0 for 푖 = 푗 = 1,2 , 푥푖 푥푗 = 푥푖 푥푗 = 0, for 푖 ≠ 푗 not bounded, a contradiction. induce a ring structure R of G with involution *- 2) Suppose that 퐻 = 푥 ∗ and 퐾 = 푒 ∗ with satisfaying 푅2 ≠ 0. Therefore there exist non zero 푥 = 푛, and 푒 = ∞. The products 푥2 = 푥푒 = integer 푘 , 푘 such that 푅 = < 푘 푥 + 푘 푥 >∗ .Every 푒푥 = 푒∗ = 푥 = 푒푥∗ = 푥푒∗ = 푥∗푒 = 0 and 1 2 1 1 2 2 푥 ∈ 푅 is of the form: 푒2 = 푛푒 induce a *- ring structure R on G 풙 = 풓풙 + ퟑ풔풙 풌 풙 + 풓풙 + ퟑ풕풙 풌 풙 + 풓′ + satisfaying 푅2 ≠ 0 . Therefore there exist ퟏ ퟏ ퟐ ퟐ 풙 ퟑ풔풙′풌ퟏ풙ퟏ∗+풓풙′+ퟑ풕풙′풌ퟐ풙ퟐ∗ where 풓풙,푟푥′,푠푥,푡푥, 푠푥′,푡푥′ integers 푠 and 푡 such that 푅 =< 푠푥 + 푡푒 >∗ . are integers. From 푟푥1 + 3푠푥1 = ±1, it follows that Every 푦 ∈ 푅 is of the form 푦 = 푚푦 푠푥 + ∗ ∗ ∗ ∗ 푟푥1 ≡ ±1(푚표푑3) . However 푟푥1 + 3푡푥1 = 0 implies 푚푦 + 푢푦 푛 + 푚푦 푠푥 + 푚푦 + 푢푦 푛 푡푒 , with 푟푥1 ≡ 0(푚표푑3) which is a contradiction. 푚 , 푚∗ , 푢 , and 푢∗ integers. In particular 푦 푦 푦 푦 Theorem1.24.Let G be torision strongly principal 푚 + 푢 푛 푡 = 1 . Hence 푡 = ±1 . 푒 푒 *- ideal ring group. Then G is a *- cyclic group or Therefore, 푚푥 + 푢푥 푛 = 0 and so 푛\푚푥 , ∗ ∗ 퐺 = 푥1 ⊕ 푥2 , with 푥푖 = 푝 , a prime , where however , 푥 = 푚 푠푥 = 0, is a contradiction. 푥 푖 = 1,2. Theorem1.22.Let G be a mixed group. Then Proof:Suppose that G is a strongly principal *- ideal ring group. Let G be indecomposable .then 1. If G is a principal *- ideal ring group, then 퐺 ≅ 푍푝푛 , where p is a prime , 1 ≤ 푛 ≤ ∞. If 푛 = ∞, 퐺푡 is bounded and 퐺\퐺푡 is a principal *- ideal ring group. then G is divisible and so G is nil, which is a contradiction. Hence G is cyclic and so *- cyclic. 2. Conversely, if 퐺푡 is bounded and if there exists a unital principal *- ideal ring with Next suppose that 퐺 = 퐻 ⊕ 퐾, 퐻 ≠ 0, 퐾 ≠ 0, either H or K are both *- cyclic or both nil. If H and K are nil, additive group 퐺\퐺푡 , then G is a principal *- ideal ring group. then they are both divisible, so G is nil which is ∗ ∗ again a contradiction. Therefore 퐺 = 푥1 ⊕ 푥2 Proof: 1) Let R be a principal *- ideal ring with with 푥푖 = 푛푖 푖 = 1,2. If 푛1, 푛2 = 1, then G is *- cyclic + ∗ 푅 = 퐺 . Since 퐺푡 a *- ideal in R, 퐺푡 =< 푥 > and . other wise, let p be a prime divisor of 푛1, 푛2 . Then ∗ ∗ 푛퐺푡 = 0, 푛 = 푥 now 퐺 = 퐺푡 ⊕ 퐻 and 퐻 ≅ 퐺\퐺푡 . Now 퐺 = 푦1 ⊕ 푦2 ⊕ 퐻 with 푦푖 = 푝푚푖 , 푖 = 1,2 and ∗ ∗ 푅 =< 푎 + 푦 >,푎 ∈ 퐺푡 , 0 ≠ 푦 ∈ 퐻. 1 ≤ 푚1 ≤ 푚2. Since 푦1 ⊕ 푦2 is neither *- cyclic 2 푚 ∗ Suppose that 푅 ⊆ 퐺푡 and let 푕 ∈ 퐻 . Then there nor nil, 퐻 = 0. The product 푦푖 푦푗 = 푝2 − 푙푦2, 푦푖 푦푗 = 0 ∗ exist integers 푘푛 , 푘푛 such that , where 푖, 푗 = 1,2 induce a *- ring structure R on G ∗ ∗ 2 2 2 ∗ 푕 = 푘푛 푦 + 푘푛 푦 + 푏 with 푏 ∈ 푅 since 푅 ⊆ 퐺푡 , 푏 = 0 with 푅 ≠ 0 . therefore, 푅 =< 푠1푦1 + 푠2푦2 > , where ∗ ∗ ∗ and 푕 = 푘푛 푦 + 푘푛 푦 . Therefore 퐻 = 푦 . H is a 푠1 푎푛푑 푠2 are integers. Every element 푥 ∈ 푅 has the principal *- ideals ring group. form : 2 푚 ′ If 푅 ⊈ 퐺푡 , then 푅 = 푅\퐺푡 is a principal *- ideal ring 푥 = 푘푥 푠1푦2 + 푘푥 푠2 + 푚푥 푝2−1 푦2 + 푘푥 푠1푦1 + with 푅+ ≅ 퐺\퐺 , and 푅2 ≠ 0. ′ 푚 ∗ 푡 푘푥 푠2 + 푚푥 푝2−1 푦2.

54 International Journal for Modern Trends in Science and Technology

P Sreenivasulu Reddy and Kassaw Benebere : Involution on rings

Preposition1.25.Let R be *- ring. Then the following are equivalent; i. H∗(R )^2 ≠ 0 ii. R is *- prime ring with at least one minimal ideal ring.

Proof:Let 퐻∗ = 퐻∗(푅) . Assume that (퐻∗)^2 ≠ 0 . Then 퐻∗ is minimal *- ideal of R. either 퐻∗ minimal of the ring R, or 퐻∗ = 퐾 ⊕ 퐾∗ where K and 퐾∗ are minimal ideals of R. in either case, if A and B are non zero *- ideals of R, then 퐻∗ ⊆ 퐴, 퐻∗ ⊆ 퐵 and hence (퐻∗)2 ≠ 0 ⊆ 퐴퐵. So R is *-prime ring. Conversely assume that (ii). Let I be a fixed minimal *-ideal of R and let A be any non zero *-ideal of R. then 퐴퐼 ≠ 0 . So 퐴 ∩ 퐼 = 퐼 . Thus 퐼 ⊆ 퐴 for each *- ideal A of R, yielding 퐻∗ = 퐼. Since *- prime we have (퐻∗)2 ≠ 0. From this we conclude that every prime ideal is maximal ideal.

REFERENCES [1] Abu Rewash,U.A,(2000) on involution rings ,East –West J.Math,Vol.2,No. 2,102-126 [2] Feigal Stoke.S,(1983) Additive groups of rings,Pitman(APP),1983. [3] Feigelstock,Z.Schlussel .S,(1978) Principal ideal and and noetherian groups, Pac.J.Math, 75,85-87. [4] Fuchs.L(1970) infinite Abelian Groups, Vol. I,Academic Press, New York. [5] Fuch.L, (1973) infinite abelian groups, Vol,II, academic press, New York. [6] Hungerford T.W,(1974) algebra, Springer science + Business media [7] Robinson D. J.S, (1991) a course in the Theory of groups, Springer, New .York

55 International Journal for Modern Trends in Science and Technology