Các kiểu căn Jacobson của các đại số đường đi Leavitt với hệ số trong nửa vành có đơn vị giao hoán

ing is 
irreducible if for an arbitrarily fixed pair of Theorem 2.3. [Katsov and Nam (2014), 
elements u,' u M with uu' and any Corollary 5.11]. For all matrix hemirings 
 M( R ), n 1, over a hemiring R, the following 
mM, there exist a,' a R such that n
 equations hold: 
 m au a''''. u au a u 
 (a) JMRMJR(nn ( )) ( ( )); 
44 
 Dong Thap University Journal of Science, Vol. 9, No. 5, 2020, 42-50 
 (b) JMRMJR( ( )) ( ( )). path if s()() f s e but fe 
 s n n s i i 
 Theorem 2.4. [Mai and Tuyen (2017), for some 1. in 
Corollary 1]. Let R be a hemiring and RR12, A graph is acyclic if it has no cycles. 
 A graph is said to be a no-exit graph if no 
be its subhemirings. If RRR 12, then 
 cycle in has an exit. 
JRJRJR()()() 12 and JRJRJRs( )  s (12 ) s ( ). 
 Remark 3.1. If E is a finite acyclic 
 3. The Leavitt path algebras 
 graph, then it is a no-exit graph, and the 
 In this section, we survey some concepts converse is not true in general. 
and results from previous works (Abrams & 
 Definition 3.2 [Katsov et al. (2017), 
Pino, 2005; Katsov et al., 2017; Abrams, 01
2015), and use them in the main section of this Definition 2.1]. Let E (,,,) E E s r be a graph 
article. First, we recall the Leavitt path and R be a commutative semiring. The Leavitt 
algebras having coefficients in an arbitrary path algebra LR ()E of the graph with 
commutative semiring. coefficients in is the R algebra presented 
 by the set of generators E0 EE1 ()1* where 
 A (directed) graph E (,,,) E01 E s r 
 E1 (),, E 1* e e * is a bijection with 
consists of two disjoint sets E0 and E1 - 
 EEE0,,() 1 1 * pairwise disjoint, satisfying the 
vertices and edges, respectively - and two 
 following relations: 
maps r,:. s E10 E If eE 1, then se and 
 vw  w 
re are called the source and range of e, (1) vw, ( is the Kronecker 
 symbol) for all v, w E0 ; 
respectively. The graph E is finite if 
E0 and E1 . A vertex vE 0 for ***
 (2) s()() e e e er e and r()() e e e e s e 
 1
which s 1 (v) is empty is called a sink; and a for all eE ; 
 1
vertex is regular if 0 s (v) . In (3) e* f  r() e for all e, f E1; 
 ef, 
this article, we consider only finite graphs. 
 (4) v ee* whenever vE 0 is 
 A path in a graph E is a  
 p e12 e... en e s 1 () v
 1
sequence of edges e12, e ,..., en E such that a regular. 
r eii s e 1 for in 1,2,..., 1. In this case, The following are two structural theorems 
we say that the path p starts at the vertex of the Leavitt path algebras over any field K
s p :() s e and ends at the vertex of acyclic graphs, no-exit graphs and 
 1 applicable examples. 
r():, en r p and has length pn . We 
 Theorem 3.3 [Abrams (2015), Theorem 
 0
consider the vertices in E to be paths of 9]. Let E be a finite acyclic graph and K any 
length 0. If s p r( p ), then is a closed 
 field. Let ww1,..., t denote the sinks of E (at 
path based at v s p r( p ). If c e12 e... en is least one sink must exist in any finite acyclic 
a closed path of positive length and all vertices graph). For each wi , let ni denote the number 
s( e12 ), s ( e ),..., s ( en ) are distinct, then the path c of elements of path in having range vertex 
is called a cycle. An edge f is an exit for a wi (this includes itself, as a path of length 
 0). Then 
 45 
Natural Sciences issue 
 t 4. Main results 
 LEMK( ) ( ).
 Kni 
 i 1 In this section, we calculate the J
 Example 3.4. Let K be a field and E a 
 radical and the J radical for the Leavitt path 
finite acyclic graph has form s
 LE()
 algebras R with coefficients in a 
 commutative semiring R of some finite 
 directed graphs E. In particular, we calculate 
 Figure 1 
 the radical and the radical for the 
 has two sinks {vv12 , }, v1 has two paths 
 Leavitt path algebras LE()with coefficients 
{,}ve having range vertex v and v has two K 
 1 1 2 in a field K of acyclic graphs, no-exit graphs 
 vf, v .
paths 2  having range vertex 2 From and applicable examples. 
Theorem 3.3, we have 
 Proposition 4.1. Let R be a commutative 
 01
 LEMKMKK ( )22 ( ) ( ). semiring and E (,,,) E E s r a graph has form 
 Theorem 3.5 [Nam and Phuc (2019), 
Corollary 2.12]. Let be a field, a finite 
no-exit graph, {cc ,..., } the set of cycles, and 
 1 l 
{vv1 ,...,k } the set of sinks. Then Figure 3 
 0 1
 kl i.e., E {}v and E {e }. Then 
 1
 LK()( E  M m 11 ())( K   M n ([, K x x ])), 
 ij 11ij 1
 J( LR ( E )) J ( R [ x , x ]) và 
 m 1
where for each 1, ik i is the number of J( L ( E )) J ( R [ x , x ]),
 s R s 
path ending in the sink vi , for each 1, jl 
 where R[,] x x 1 is a Laurent polynomials 
n j is the number of path ending in a fixed 
 algebra over semiring R. 
(although arbitrary) vertex of the cycle c j 
which do not contain the cycle itself and Proof. It is well known that 
 1 *
K[,] x x Laurent polynomials algebra over LR (),, E R v e e is a Leavitt path algebra 
field K. generated by set {,,}v e e* and Laurent 
 Example 3.6. Let be a field and a polynomials algebra R[,] x x 1 generated by 
finite no-exit graph has form 
 set {xx , 1 }. Consider the map 
 f:()[,] L E R x x 1
 R 
 determined by fv( ) 1, f() e x and 
 Figure 2 
 f(). e*1 x Then, it is easy to check that f is 
 has only one cycle e0 , no sink and one path 
 an algebraic isomorphism, i.e., 
e1 other cycle e0 having range vertex v0 . 
 L( E ) R [ x , x 1 ],
From Theorem 3.5 deduced R 
 L( E ) M ( K [ x , x 1 ]). the proof is completed. □ 
 K 2 
 Proposition 4.2. Let be a commutative 
 Remark 3.7. From Remark 3.1, Theorem 
3.3 is a corollary of Theorem 3.5. semiring and a graph has form 
46 
 Dong Thap University Journal of Science, Vol. 9, No. 5, 2020, 42-50 
 Leavitt path algebra generated by set 
 v,..., v , e ,..., e , e** ,..., e
 1n 1 n 1 1 n 1 and 
 M( R ) R E |1 i , j n , 
 n i, j
 Figure 4 is a matrix algebra generated by set 
 E0 {}v E1 {ee ,..., } n 1.
 i.e., and 1 n with E|1 i , j n , where E are the standard 
 ij,  ij,
Then 
 elementary matrices in the matrix semiring 
 JLEJLR(Rn ( )) (1, ( )) and MRn ( ). Consider the map 
 JLEJLRs( R ( )) s (1, n ( )), 
 f:()() LRn E M R 
where LR1,n () is a Leavitt algrbra type 1,n . 
 determined by f() vi E i, i , f() ei E i,1 i and 
 *
 Proof. It is well known that f() ei E i 1, i for each 1. in Then, it is easy to 
 **
LR( E ) R v , e11 ,..., e n , e ,..., e n is a Leavitt path check that is an algebraic isomorphism, i.e., 
 ** LEMRRn( ) ( ). Thence inferred 
algebra generated by set v, e11 ,..., enn , e ,..., e  
 JLEJMR( ( )) ( ( )) and JLEJMR( ( )) ( ( )).
and L( R ) R x ,..., x , y ,..., y , where Rn s R s n 
 1,n 1 n 1 n From Theorem 2.3, the proof is completed. □ 
 n
xyi j  ij and  xyii 1 for 1 i , j n , is a 
 i 1 Proposition 4.4. Let be a commutative 
Leavitt algebra type Consider the map semiring and a graph has form 
 f:()() LRn E L1, R 
Determined by fv( ) 1, f() eii x and 
f() e* y 1. in
 ii for each Then, it is easy to 
check that f is an algebraic isomorphism, i.e., Figure 6 
 0
LELRRn( ) 1, ( ), the proof is completed. □ i.e., E { v , w ,..., w } and with 
 1 n 1 
 n 2. Then and 
 Proposition 4.3. Let R be a commutative 
 01 JLEMJR( ( )) ( ( )), where is a matrix 
semiring and E (,,,) E E s r a graph has form s R n s 
 algebra over semiring 
 Proof. It is well-known that 
 **
 LR( E ) R v , w1 ,..., w n 1 , e 1 ,..., e n 1 , e 1 ,..., e n 1 is 
 Figure 5 
 a Leavitt path algebra generated by set 
 0 1
 i.e., E {vv1 ,...,n } and E {ee1 ,...,n 1 } with **
 v, w1 ,..., wn 1 , e 1 ,..., e n 1 , e 1 ,..., e n 1 . Consider 
n 2. Then 
 the map 
 JLEMJR(Rn ( )) ( ( )) và JLEMJRs( R ( )) n ( s ( )), 
 f:()() LRn E M R 
 where MRn () is a matrix algebra over 
 determined by f() v E1,1 , f() wi E i 1, i 1 , 
semiring R. *
 f() ei E i, n and f() ei E n, i for each 
 Proof. It is well-known that 
 1 in 1. Then, it is easy to check that is 
 **
LR( E ) R v1 ,..., v n , e 1 ,..., e n 1 , e 1 ,..., e n 1 is a 
 an algebraic isomorphism, i.e., LEMRRn( ) ( ). 
 Thence it infers 
 47 
Natural Sciences issue 
JLEJMR(Rn ( )) ( ( )) and JLEJMRs( R ( )) s ( n ( )). ending in the sink vi , for each 1, jl is 
From Theorem 2.3, the proof is completed. □ the number of path ending in a fixed (although 
 arbitrary) vertex of the cycle which do not 
 Corollary 4.5. Let R be a commutative 
 contain the cycle itself. 
semiring and E (,,,) E01 E s r a graph has form 
Figure 5 or Figure 6. Then From Theorem 2.4, we have 
 kl
 (a) If R then JLEJLE( ( )) ( ( )) 0, 1
 s JLE(K ( )) (  JM ( m 11 ( K )))  (  JM ( n ( Kxx [ , ]))),
 ij 11ij
where is the commutative semiring of non- kl
 1
 JLEs( K ( )) (  JM s ( m 11 ( K )))  (  JMKxx s ( n ( [ , ]))).
negative integers. ij 11ij
 (b) If R be a unita commutative ring, then From Theorem 2.3, we have 
JLEJLEMJR(R ( )) s ( R ( )) n ( ( )), where JR() is kl
 1
 JLE(K ( )) (  M m 11 ( JK ( )))  (  M n ( JKxx ( [ , ]))),
a Jacobson radical of ring R. ij 11ij
 kl
 1
 (c) If K is a field, then JLEs( K ( )) (  M m 11 ( JK s ( )))  (  MJKxx n ( s ( [ , ]))).
 ij 11ij
JLEJLE(K ( )) s ( K ( )) 0. 
 From K is a field and Remark 2.2, we have 
 Proof. (a) According to Lemma 5.10 of 
 JKJK( ) s ( ) 0, the proof is completed. □ 
Katsov and Nam (2014), JJ( ) s ( ) 0. 
 Example 4.7. (a) Let be field and a 
 (b) Since R is a ring, JRJR( ) s ( ). graph has form Figure 3. Since graph in 
 (c) Since K is a field, Figure 3 is no-exit, there exists only one cycle 
 e, no sink and not path other cycle e having 
 JKJK( ) ( ) 0. 
 s ending in vertex v. From Theorem 4.6, we 
 From Proposition 4.3 or Proposition 4.4, 1
 have J( LK ( E )) J ( K [ x , x ]) and 
the proof is completed. □ 
 J( L ( E )) J ( K [ x , x 1 ]).
 Theorem 4.6. Let K be an any field, E a s K s 
finite no-exit graph, {cc1 ,...,l } the set of cycles, This result is also the result in Proposition 
 4.1 when the commutative semiring R is a field. 
and {vv1 ,...,k } the set of sinks. Then 
 l (b) Let be a field and a graph has 
 1 
 (a) J( LKn ( E ))  M 1 ( J ( K [ x , x ])), 
 j 1 j form Figure 4. Since graph in Figure 4 is no-
 l exit, there is n cycles e j for each 1, jn no 
 1 
 (b) Js( L K ( E ))  M n 1 ( J s ( K [ x , x ])), 
 j 1 j sink and for each 1, jn has n 1 paths 
 other cycle e j having ending vertex v in cycle 
where for each 1, jl n j is the number of 
path ending in a fixed (although arbitrary) e j . From Theorem 4.6, we have 
 11
vertex of the cycle c j which do not contain the JLE(K ( )) MJKxx n ( ( [ , ]))  ...  MJKxx n ( ( [ , ])),
 1 11
cycle itself and K[,] x x Laurent polynomial JLEs( K ( )) MJKxx n ( s ( [ , ]))  ...  MJKxx n ( s ( [ , ])),
algebra over field K. the directed sum of the right hand side has n 
 terms. This result is also the result in 
Proof. From Theorem 3.5, we have 
 Proposition 4.2 when the commutative 
 kl
 1 semiring is a field, because 
 LK()( E  M m 11 ())( K   M n ([, K x x ])),
 ij 11ij 
 11
 L1,n( K ) M n ( K [ x , x ])  ...  M n ( K [ x , x ]). 
where the set of cycles, and 
the set of sinks for each 1, ik m is of path (c) Let be a field and be a no-exit 
 i graph has form Figure 2. From Theorem 4.6, 
48 
 Dong Thap University Journal of Science, Vol. 9, No. 5, 2020, 42-50 
we have J( L ()) E M (([, J K x x 1 ])) and LE() with coefficients in a commutative 
 K 2 R 
J( L ( E )) M ( J ( K [ x , x 1 ])). semiring R of some finite graphs E
 s K2 s 
 (Proposition 4.1, Proposition 4.2, Proposition 
 Corollary 4.8. Let K be a any field, E a 4.3, Proposition 4.4). In particular, we have 
finite no-cycle graph and {vv ,..., } the set of 
 1 k also calculated the radical and the 
sinks. Then 
 radical for the Leavitt path algebras LEK ()
 JLEJLE( ( )) ( ( )) 0. 
 K s K with coefficients in a field K of acyclic graphs 
 Proof. It immediately follows from (Corollary 4.8), no-exit graphs (Theorem 4.6) 
Theorem 4.6. □ and applicable examples (Example 4.7 and 
 Example 4.10). 
 Remark 4.9. We can use Theorem 3.3 to 
proof Corollary 4.8. Especially, from Theorem In the future, we will expand two 
3.3 we have structural theorems (Theorem 3.3 and Theorem 
 3.5) of the Leavitt path algebras over 
 t
 LEMK( ) ( ), commutative semirings of acyclic graphs and 
 Kni
 i 1 no-exit graphs. 
where {ww ,..., } the set of sinks for each 
 1 t Acknowledgments: This article is 
1, itn
 i is the number of path ending in the partially supported by lecturer project under 
sink wi (this includes wi itself, as a path of the grant number SPD2017.01.27 in Dong 
length 0). Thap University./. 
 Fom Theorem 2.4, we have References 
 t t G. Abrams. (2015). Leavitt path algebras: the 
JLEJMK(Kn ( ))  ( ( )), JLEJMKs( K ( ))  s ( n ( )). 
 i 1 i i 1 i first decade. Bulletin of Mathematical 
 Fom Theorem 2.3, we have Sciences, (5), 59-120. 
 t t G. Abrams and G. Aranda Pino. (2005). The 
JLEMJK(Kn ( ))  ( ( )), JLEMJKs( K ( ))  n ( s ( )).
 i 1 i i 1 i Leavitt path algebra of a graph. Journal of 
 Algebra, (293), 319-334. 
 From Corollary 2.2, JKJK( ) s ( ) 0. We 
have S. Bourne. (1951). The Jacobson radical of a 
 semiring. Proceedings of the National 
 Example 4.10. (a) Let be a field and Academy of Sciences of the United States 
 a graph has form Figure 5 or Figure 6. of America, (37), 163-170. 
Since Figure 5 or Figure 6 graphs is acyclic, 
 J. Golan. (1999). Semirings and their 
follow Corollary 4.8 JLEJLE( ( )) ( ( )) 0. 
 K s K Applications. Kluwer Academic 
This is also the result in Corollary 4.5 (c). Publishers, Dordrecht-Boston-London. 
 (b) Let be is a field and a acyclic K. Iizuka. (1959). On the Jacobson radical of a 
graph has form in Example 3.4. From semiring. Tohoku Mathematical Journal, 
Corollary 4.8, (11), 409-421. 
 JLEJLE( ( )) ( ( )) 0.
 K s K Y. Katsov and T. G. Nam. (2014). On radicals 
 5. Conclusion of semirings and ralated problems. 
 Communication Algebra, (42), 5065-5099. 
 We have calculated the J radical and 
the Js radical for the Leavitt path algebras 
 49 
Natural Sciences issue 
W. G. Leavitt. (1962). The module type of a Semirings and Related Problems. Vietnam 
 ring. Transactions of the American Journal of Mathematics, (45), 493-506. 
 Mathematical Society, (103), 113-130. T. G. Nam and N. T. Phuc. (2019). The 
Y. Katsov, T. G. Nam and J. Zumbrägel. structure of Leavitt path algebras and the 
 (2017). Simpleness of Leavitt path Invariant Basis Number property. Journal 
 algebras with coefficients in a of Pure and Applied Algebra, (223), 4827-
 commutative semiring. Semigroup Forum, 4856. 
 (94), 481-499. M. Tomforde. (2011). Leavitt path algebras 
L. H. Mai and N. X. Tuyen. (2017). Some with coefficients in a commutative ring. 
 remarks on the Jacobson Radical Types of Journal of Pure and Applied Algebra, 
 (215), 471-484. 
50