ON THE ADDITION OF UNITS AND NON-UNITS IN FINITE COMMUTATIVE RINGS

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1 ROCKY MOUNTAIN JOURNAL OF MATHEMATICS Volume 45, Number 6, 2015 ON THE ADDITION OF UNITS AND NON-UNITS IN FINITE COMMUTATIVE RINGS DARIUSH KIANI AND MOHSEN MOLLAHAJIAGHAEI ABSTRACT. Le R be a fne commuave rng. In hs paper, we fnd he number of represenaons of a fxed member of R o be he sum of k uns n R, and he sum of k non-uns, and as a sum of a un and a non-un. We prove ha, f Z 2 s no a quoen of R, hen every r R can be wren as a sum of k uns, for each neger k > Inroducon and prelmnares. The sudy of algebrac srucures usng he properes of graphs has become an excng research opc n he las hry years, leadng o many fascnang resuls and quesons. There are many papers on assgnng a graph o a rng, see for example, [1, 2, 6, 7, 8]. Ineres n he queson of how he uns of a rng mgh generae he rng addvely goes back o he mddle of he las cenury. In 1958, Skornyakov [11, Page 167, Problem 31] posed he problem of deermnng whch regular rngs are generaed by her uns. More precsely, he asked: Is every elemen of a Von Neumann regular rng, whch canno have Z 2 as a quoen, a sum of uns? In 1974, Raphael [9] launched a sysemac sudy of rngs generaed by her uns, whch he called S-rngs. Fnally, n 1976, Fsher and Snder [5] proved ha, f R s a Von Neumann regular rng wh arnan prmve facor rngs n whch 2 s a un, hen every elemen of R can be expressed as he sum of wo uns AMS Mahemacs subjec classfcaon. Prmary 05C50, 05C25. Keywords and phrases. Unary Cayley graph, walks, un, non-un. The research of he frs auhor was n par suppored by a gran of IPM (gran No ). The frs auhor s he correspondng auhor. Receved by he edors on May 15, 2013, and n revsed form on December 11, DOI: /RMJ Copyrgh c 2015 Rocky Mounan Mahemacs Consorum 1887

2 1888 DARIUSH KIANI AND MOHSEN MOLLAHAJIAGHAEI Theorem 1.1. [9]. Le R be a fne commuave rng wh nonzero deny. Then R s generaed by s uns f and only f R canno have Z 2 Z 2 as a quoen. In [10], he auhors found he number of represenaons of a fxed resdue class mod m as he sum of wo uns n Z m, he sum of wo non-uns, and he sum of mxed pars, respecvely. We generalze hese resuls no commuave rngs, as he sum of k uns and he sum of k non-uns. In addon, we prove Theorem 1.1. Our mehods are dfferen and much smpler. Throughou hs paper, R s a fne commuave rng wh deny. We denoe he se of un elemens by R. We say ha R s local f R has exacly one maxmal deal. If R s a fne commuave rng, hen R R 1 R, where each R s a fne commuave local rng wh maxmal deal M, by [3, Theorem 8.7]. I s obvous ha R/J R R 1 /M 1 R /M, where J R s he Jacobson radcal of R, snce he Jacobson radcal s he produc of he maxmal deals of he local facors. Ths decomposon s unque up o permuaon of facors. We denoe by k he (fne) resdue feld R /M, π : R k he quoen map, and f = k. We also assume (afer an approprae permuaon of facors) ha f 1 f 2 f. Clearly, (u 1,..., u ) s a un of R f and only f each u s a un elemen n R for = 1,...,. The movaon of hs paper s he deermnaon of he number of soluons of (1.1) x 1 + x x k = r where x are eher all uns or all non-uns. Le A be an addve group wh deny 0. For S A such ha S = { s; s S} = S he Cayley graph X = Cay (A, S) s he undreced graph havng verex se V (X) = A and edge se E(X) = {{a, b}; a b S}. Clearly, f 0 / S, hen here s no loop n X, and f 0 S, hen here s exacly one loop a each verex. I s easy o check ha S generaes A f and only f X s conneced. The unary Cayley graph of a rng R, denoed by G R, s he graph whose verex se s R, and n whch {x, y} s an edge f and only f x and y are elemens of R such ha x y R. Le G be a graph. Suppose v 0 and v n are wo verces of G. A

3 FINITE COMMUTATIVE RING UNITS AND NON-UNITS 1889 (v 0, v n )-walk n G s a sequence W := v 0 e 0 v 1 e 1 v 2 e n 2 v n 1 e n 1 v n, whose erms are alernaely verces and edges of G (no necessarly dsnc), such ha v 1 and v are he endpons of e for all. We denoe by w k (G, x, y) he number of x y walks of lengh k n he graph G. I s neresng o fnd he number of walks n Cay (R, R ). The ensor produc G 1 G 2 of wo graphs G 1 and G 2 s he graph wh verex se V (G 1 G 2 ) := V (G 1 ) V (G 2 ), wh edges specfed by pung (u, v) adjacen o (u, v ) f and only f u s adjacen o u n G 1 and v s adjacen o v n G 2. So, we mmedaely see ha G R s he ensor produc of he graphs G R1,..., G R. Le A G be he adjacency marx of he smple graph G. The nex lemma can be proved easly by nducon. Lemma 1.2. [13]. If A G s he adjacency marx of a graph G n such a way ha he verces are labeled by v 1, v 2,..., v n, hen he (, j) enry of A k, s he number of v v j walks of lengh k n G. We refer he reader o [13] for general defnons of graph heory. The nex heorem deermnes he dameer of unary Cayley graphs. Theorem 1.3. [2]. If R = R 1 R s a produc of local rngs, hen 1 f = 1 and R s a feld; 2 f = 1 and R s no a feld; dam (G R ) = 2 f > 1, f 1 > 2; 3 f > 1, f 1 = 2, f 2 > 2; f > 1, f 1 = f 2 = 2. The above resul generalzes Theorem 1.1, snce G R s a dsconneced graph f and only f > 1 and f 1 = f 2 = 2. In he oher cases, G R s conneced and R s generaed (addvely) by s uns. 2. Represenaon by uns. The purpose of hs secon s o fnd he number of represenaons of a member of rng R as he sum of k uns of R.

4 1890 DARIUSH KIANI AND MOHSEN MOLLAHAJIAGHAEI Lemma 2.1. [2]. Le R be a fne commuave rng. () G R s a regular graph of degree R. () If R = R 1 R s a produc of local rngs, hen G R = =1 G R. () If R s a commuave local rng wh maxmal deal M, hen G R s a complee mulpare graph whose pare ses are he coses of M. We denoe he number of soluons of equaon (1.1) n R R R, by S u (R, r, k) and S nu (R, r, k), respecvely. Nex are some basc consequences of hs defnon. and Lemma 2.2. Le R be a rng, and r be an arbrary member of R. Le k be a naural number. Then he followng hold: (a) S u (R, r, k) = w k (Cay (R, R ), r, 0). (b) S nu (R, r, k) = w k (Cay (R, R R ), r, 0). (c) If r s a un of R, hen S u (R, r, k) = S u (R, r r, k), and S nu (R, r, k) = S nu (R, r r, k). Proof. (a) Le S and W be he se of soluons of equaon (1.1) n R and he se of walks of lengh k beween 0 and r, respecvely. Le φ : S W be defned as φ((x 1, x 2,..., x k )) = (0, x 1, x 1 + x 2,..., x x k ). Obvously, φ s one o one and ono. So, S u (R, r, k) = w k (Cay (R, R ), r, 0). (b) The proof s smlar o (a). (c) Le S r and S r r be he se of soluons of equaon (1.1) n R, for r and r r, respecvely. Le ψ : S r S r r be defned as ψ((x 1, x 2,..., x k )) = (r x 1, r x 2,..., r x k ). Obvously, φ s one o one and ono. So, S u (R, r, k) = S u (R, r r, k). Applyng he same argumen as above we ge S nu (R, r, k) = S nu (R, r r, k).

5 FINITE COMMUTATIVE RING UNITS AND NON-UNITS 1891 In wha follows, we generalze Theorems 1.1 and 1.3. Our proofs are based on walks on unary Cayley graphs. We need he followng resul, whose proof s clear from he defnon of adjacency n he ensor produc. Lemma 2.3. Le G and H be smple graphs. Then w k (G H, (x 1, y 1 ), (x 2, y 2 )) = w k (G, x 1, x 2 )w k (H, y 1, y 2 ). Theorem 2.4. Le R be a fne local rng wh maxmal deal M of sze m. Then we have he followng: S u (R, r, k) = where = R /m. {( ( 1)k ( 1) k + ( 1) k )m k 1 f r M ( 1) k ( 1) k m k 1 f r R M, Proof. By Lemma 2.1, he graph G R s a complee mulpare graph, so he adjacency marx of G R s equal o A G = (J I ) J m, where J m s he m m all 1-marx. Thus, by he fac ha, for arbrary marces A, B, C and D, we have (A B)(C D) = (AC BD), so: ( k ( ) k (2.1) A k G = (( 1) ) k 1 J ) + ( 1) k I (m k 1 J m ). =1 We can easly see ha k ( k (( 1) ) k 1 ) = ( 1)k ( 1) k. =1 Therefore, f r M, hen ( ( 1) k ( 1) k S u (R, r, k) = + ( 1) )m k k 1 ; oherwse, S u (R, r, k) = ( 1)k ( 1) k m k 1. The nex heorem, whch can be easly obaned by Lemma 2.3 and Theorem 2.4, gves he number of soluons of equaon (1.1) for a fxed member of R.

6 1892 DARIUSH KIANI AND MOHSEN MOLLAHAJIAGHAEI Theorem 2.5. Le R be a rng, where R = R 1 R 2 R and R s a local rng wh maxmal deal M of sze m, for all {1, 2,..., }. Le r = (r 1,..., r ) be a member of R. Then (( ( 1) k ( 1) k ) ) a (2.2) S u (R, r, k) = + ( 1) k m k 1 =1 ( (( 1) k ( 1) k )m k 1 ) 1 a, where = R /m and a = 1 f and only f r M, oherwse a = 0. Corollary 2.6. A rng R s generaed by uns f and only f R does no have Z 2 Z 2 as a quoen. I s enough o check ha, for r R, w k (G R, r, 0) s no zero. Remark 2.7. I should be menoned ha Theorem 1.3 can be derved from Theorem 2.5. Theorem 2.5 s a generalzaon of [10, Theorem 1.1] and [4, Theorem 4.4]. In he followng, we sudy he good number of elemens of a rng. Defnon 2.8. [12]. An elemen r R s called k-good f r = u 1 + u u k wh u 1, u 2,..., u k R, and he rng R s called k-good f every elemen of R s k-good. The rng R s called ω-rng f s no k-good for any k, bu every elemen of R s k-good for some k (ha s, when a leas R generaes R addvely); oherwse, R s called an -rng. Clearly, a un elemen s 1-good. The good number of a rng has been suded n [12]. In he sequel, we prove ha, f R s a rng where Z 2 s no a quoen of R, hen R s a k-good rng, for each k > 1. Oherwse, R s no 2-good. Theorem 2.9. Le R be a rng and k > 1 a naural number. Then R s k-good f and only f Z 2 s no a quoen of R. Proof. Le R be a rng such ha Z 2 s no a quoen of R. defnon, s enough o prove ha each r R s k-good. By By

7 FINITE COMMUTATIVE RING UNITS AND NON-UNITS 1893 Theorem 2.5, s enough o show ha (2.3) (( ( 1) k ( 1) k =1 To show hs, we prove ha: ( ( 1) k ( 1) k (2.4) and These nequales are equvalen o: (2.5) and f and only f ) ) a + ( 1) k m k 1 ( (( 1) k ( 1) k )m k 1 ) 1 a 0. ) + ( 1) k m k 1 0 (( 1) k ( 1) k )m k 1 0. ( 1) k ( 1) k + ( 1) k 0 ( 1) k ( 1) k 0, (2.6) ( 1) k 1 ( 1) k and ( 1) k ( 1) k, f and only f 0, 2 for all. The proof of he nex resul s smlar. Theorem Le R be a rng. Then R s an ω-rng f and only f Z 2 Z 2 s no a quoen of R, bu Z 2 s a quoen of R. Remark Le R be a rng. Z 2 Z 2 s a quoen of R. Then R s an -rng f and only 3. Represenaon by non-uns. In hs secon, we sudy hose rngs whch are generaed by her non-un members. In wha follows,

8 1894 DARIUSH KIANI AND MOHSEN MOLLAHAJIAGHAEI we fnd he number of soluons of he followng equaon n non-un members of a rng R: (3.1) x 1 + x x k = r. Obvously, f R s a local rng, hen here s no soluon o he above equaon for un members. The nex heorem shows he relaon beween S nu (R, r, k) and S u (R, r, k): Theorem 3.1. Le R = R 1 R 2 R s be a fne rng where, for {1, 2,..., s}, each R s local of order n wh maxmal deal M of order m. Le n = s =1 n. Then he number of soluons of (3.1) n whch all x are non-uns s gven by: (3.2) S nu (R, r, k) = ( 1) k S u (R, r, k) k ( ( ) ( s k + ( 1) k n 1 (n m ) )). k =1 =1 Proof. We assgn labels o he verces of G R such ha v 1 = 0 and v 2 = r. We know ha J n I n A GR s he adjacency marx of he complemen graph G c R. The adjacency marx of Cay (R, R R ) s (J n I n A G ) + I n, snce G c R s somorphc o he Cay (R, R (R {0})). By Lemma 2.2, S nu (R, r, k) s he (1, 2) enry of marx (J n I n A G + I n ) k = (J n A G ) k. So, (3.3) S nu (R, r, k) = ((J n A G ) k ) 1,2 ( k ( ) ) k = ( 1) k (J n ) A k G. 1,2 =0 Snce G R s an s =1 (n m )-regular graph, follows ha and J n A k G s = (n m ) k J n =1 (J n ) = n 1 J n.

9 FINITE COMMUTATIVE RING UNITS AND NON-UNITS 1895 Therefore, we have: (3.4) S nu (R, r, k) = Thus, ( ( 1) k A k G + k ( ) k ( 1) k n 1 ( s ) k ) J n =1 =1 (n m ). 1,2 (3.5) S nu (R, r, k) = ( 1) k S u (R, r, k) k ( ( ) ( s k + ( 1) k n 1 (n m ) ). k =1 =1 Remark 3.2. Le R be a fne local rng wh maxmal deal M of sze m. Then, for he number of soluons of equaon (3.1) n non-un members of R, we have he followng: (a) If r R M, hen here s no soluon. (b) If r M, hen hs number s m k 1. The followng corollary s obaned from Theorems 2.4 and 3.1. Corollary 3.3. Le R be a fne rng. Then he number of soluons of he equaon (3.6) x 1 + x 2 = r, where x 1 R and x 2 R R s (3.7) R S u (R, r, 2) S nu (R, r, 2). 2 I would be neresng o calculae he number of soluons of equaon (1.1) for mxed pars, and we leave as an open problem. Acknowledgmens. The auhors acknowledge he careful readng and excellen suggesons of he anonymous referees.

10 1896 DARIUSH KIANI AND MOHSEN MOLLAHAJIAGHAEI REFERENCES 1. S. Akbar, D. Kan, F. Mohammad and S. Morad, The oal graph and regular graph of a commuave rng, J. Pure Appl. Alg. 213 (2009), R. Akhar, M. Boggess, T. Jackson-Henderson, I. Jménez, R. Karpman, A. Knzel and D. Prkn, On he unary Cayley graph of a fne rng, The Elecr. J. Comb. 16 (2009), #R M.F. Ayah and I.G. Macdonald, Inroducon o commuave algebra, Addson-Wesley Publshng Co, Readng, Massachuses, E. Cancela, D.A. Jaume, A. Pasne and D. Vdela, Walks on unary Cayley graphs and applcaons, avalable from: arxv: v1 [mah.co], 12 Mar Joe W. Fsher and Rober L. Snder, Rngs generaed by her uns, J. Alg. 42 (1976), D. Kan and M. Molla Haj Aghae, On he unary Cayley graph of a rng, The Elecr. J. Comb. 19 (2012), #P D. Kan, M. Molla Haj Aghae, Y. Meemark and B. Sunornpoch, Energy of unary cayley graphs and gcd-graphs, Lnear Alg. Appl. 435 (2011), H.R. Maman, M.R. Pournak and S. Yassem, Necessary and suffcen condons for un graphs o be Hamlonan, Pacfc J. Mah. 249 (2011), R. Raphael, Rngs whch are generaed by her uns, J. Alg. 28 (1974), J.W. Sander, On he addon of uns and nonuns mod m, J. Num. Theor. 129 (2009), L.A. Skornyakov, Complemened modular laces and regular rngs, Olver and Boyd, Ednburgh, Peer Vamos, 2-Good rngs, Quar J. Mah. Oxford 56 (2005), D.B. Wes, Inroducon o graph heory, Second edon, Prence-Hall, Upper Saddle Rver, New Jersey, Deparmen of Mahemacs and Compuer Scence, Amrkabr Unversy of Technology (Tehran Polyechnc), 424, Hafez Ave., Tehran 15914, Iran and School of Mahemacs, Insue for Research n Fundamenal Scences (IPM), P.O. Box , Tehran, Iran Emal address: dkan@au.ac.r, dkan7@gmal.com Deparmen of Mahemacs, Unversy of Wesern Onaro, London, ON N6A 5B7, Canada Emal address: mmollaha@uwo.ca