<span class="mathjax-formula">${\mathbb {Z}}_{k+l}\times {\mathbb {Z}}_2$</span>-graded polynomial identities for <span class="mathjax-formula">$M_{k,l}(E)\otimes E$</span>

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Z

_k₁_l

3 Z

₂

-Graded Polynomial Identities for M

_k_,_l

(E) 7 E.

ONOFRIO MARIO DI VINCENZO(*) - VINCENZO NARDOZZA(**)

ABSTRACT- LetKbe a field of characteristic zero, andEbe the Grassmann algebra over an infinite-dimensionalK-vector space. We endowM_k,_l(E)7Ewith aZ_k₁_l3Z₂-grading, and determine a generating set for the ideal of its graded polynomial identities. As a consequence, we prove that M_k,_l(E)7Eand M_k₁_l(E) are PI-equivalent with respect to this grading. In particular, this leads to their ordinary PI-equivalence, a classical result obtained by Kemer.

1. Introduction.

LetKbe a field of characteristic zero, andEbe the Grassmann algebra over an infinite-dimensional K-vector space. For fixed integers k,l (kFl) we consider the K-algebra M_k,_l(E), whose elements are the following block matrices with entries in the even and odd part ofE, resp.

E0 and E1:

g

^EE⁰1 ^EE¹0

h

k l

N

^–^––

k

N

l

l l l

As follows by the results of Kemer [K], these algebras generate non- trivial prime varieties, and their study is essential in the theory of PI-algebras. SinceMk,l(E) is a subalgebra ofMk1l(E), the following inclusion

(*) Indirizzo dell’A.: Dipartimento Interuniversitario di Matematica, via Orabona 4, 70125 Bari, Italia; e-mail: divincenzoHdm.uniba.it

(**) Indirizzo dell’A.: Dipartimento di Matematica, c.da Macchia Romana, 85100 Potenza, Italia; e-mail: vickkkHtiscalinet.it

Partially supported by MURST COFIN ’99.

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for the ideals of polynomial identities follows: T(M_k,_l(E) )*T(M_k₁_l(E) ).

It is somehow surprising that to get a PI-equivalence with M_k₁_l(E) it suffices to consider the tensor product Mk,l(E)7E, regardless to k,l, i.e. theT-ideals of the polynomial identities of these algebras are equal.

Originally, this fact was proved by Kemer in [K1] as a consequence of his structure theory for varieties of algebras. Other proofs are in the papers of Regev [R] and Berele [B]. In this paper, we shall studyM_k,_l(E)7Eas a graded algebra. Recall briefly that, for a given groupG, aK-algebraR isG-graded if, for eachgG, there is a subspaceR^gofR(theg-homoge- neous component of R) such that

R4

!

gG

R^g and R^gR^h’R^g¹^h for all g,hG.

We shall write¯_G(r)4g(or simply¯(r)4gifGis clear from the con- text) to denote theG-homogeneous degree of the homogeneous element rR^g.

The study of graded algebras is almost a standard approach in many problems of PI-theory, and many algebras have natural grading which enrich them with nice structure properties. The algebras M_n(K), M_k,_l(E), M_n(E), for instance, are Z₂-graded algebras in a natural way.

Before getting into details in the next section, we briefly recall some terminology:

LetGbe a group; for eachgGletX^gbe a countable set of non-com- muting variables, and let X^G be their disjoint union. Then the algebra KaX^Gb is a free object in the class of G-graded algebras. A polynomial f4f(x₁^g¹,R,x_r^g^r) with variablesx_i^gⁱX^gⁱis a graded polynomial identity for R if for all substitutions xig_i

Ka_iR^gⁱ (i41 ,R,r) it results f(a1,R,ar)40 . The set of all graded polynomial identities for R is an ideal of KaX^Gbinvariant under all endomorphisms of KaX^Gbpreserving the homogeneous components; we call it theT_G-ideal ofR, and denote it by T_G(R). Now call:

V_r^G»4span_Kax_{s( 1 )}^g¹ Rx_s(r)^g^r NsS_r,g₁,R,g_rGb.

We callV_r^G the space of graded multilinear polynomials, and it is easily seen that the usual left action ofSr endowsVrGwith the structure of left Sr-module as in the ordinary case. Moreover, since the fieldKis of characteristic zero, standard arguments yield thatT_G(R) is generated by its multilinear parts, i.e. by the S_r-submodules V_r^GOT_G(R) for all rN. There are many more examples of these and other concepts related to

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graded algebras; for shortness, we introduce those who are related to this paper. The first is the natural Zn-grading for the algebra Mn(K):

(M_n(K) )^t»4span_Kae_ijNj2i4tZ_nb.

Vasilovsky in [V] proved that itsT_Z_n-ideal is generated by the following multilinear polynomials:

[x₁⁰,x₂⁰] x₁^tx²^tx₂^t2x₂^tx²^tx₁^t (tZ_n).

The second instance is about the algebra M_n(E)`M_n(K)7E, which has the natural Zn3Z2-grading

(Mn(E) )^(t,^l)»4Mn(K)^t7E_l

where the first component is thet-homogeneous component ofM_n(K) in the previous grading forM_n(K). The authors in [DVN] found a system of generators for the T_Z_n₃_Z₂-ideal of its graded polynomial identities.

In this paper, for n4k1l, we define a Zn3Z2-grading for Mk,l(E)7E and describe a set of generators forTZ_n3Z₂(M_k,_l(E)7E).

In particular it turns out that this set generatesT_Z_n_3Z₂(M_n(E) ) as well.

Hence M_k,_l(E)7E and M_n(E) are equivalent as graded PI-algebras.

General arguments lead to their ordinary PI-equivalence, and we obtain a new proof for the mentioned result of Kemer, using only elementary tools.

2. Preliminaries.

Consider the K-algebra M_k,_l(E), and let n»4k1l in the following.

We may start from the naturalZ_n-grading onM_n(K) in order to endow M_k,_l(E) with the following Zn3Z2-grading:

(Mk,l(E) )^(t,^l)»4span_KaE_le_ijNj2i4tZ_nb OMk,l(E).

Of course some of the graded components may be trivial (for instance, (M_k,_l(E) )^{( 0 , 1 )}40). It is easy to verify, however, that this is actually a grading for Mk,l(E). Next, consider Mk,l(E)7E and define

(M_k_,_l(E)7E)^(t,^l)»4(M_k,_l(E) )^(t,^l)7E₀5(M_k,_l(E) )^(t^,^l¹^{1 )}7E₁. ThenM_k,_l(E)7EisZn3Z2-graded, and we shall prove that it is PI-

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equivalent to the algebra M_n(E) with the Z_n3Z₂-grading (M_n(E) )^(t,^l)4span_Kae_ij7E_lNj2i4tZ_nb.

In order to have a clearer view of the problem, the following considerations are useful:

DEFINITION 2.1. Let e:]1 ,R,n( 3 ]1 ,R,n( KZ₂ be the map defined via

e(i,j)»4^.^/

´ 0 1

if i,jGk or i,jDk otherwise .

Moreover, letE0be the naturalK-basis forE₀, andE1be the corresponding basis for E₁.

It is immediate to see that

A^»_{4 ]}aeijNi,jGn, aEe(i,j)( is a K-basis for M_k,_l(E), and

B^»4 ]ae_ij7bNi,jGn, aEe(i,j), bEl(

is a K-basis for Mk,l(E)7E. Moreover, writing a^l as a shorthand for aEl, it holds:

(M_k,_l(E) )^(t^,^l)4span_Kaa^e(i,^j)e_ijNj2i4tZn, e(i,j)4lb and

(M_k_,_l(E)7E)^(t,^l)4span_Kaa^e(i,^j)e_ij7b^l¹^e(i^,^j)Nj2i4tZ_nb. By use of these definitions, the fact that M_k,_l(E)7E is a Z_n3Z₂- graded algebra follows easily. By the way, we find useful to remark a couple of lemmas which will be of help in the following of this part.

LEMMA 2.2. Let

A_s»4ase(i_s,js)e_i_s_j_s7b_s^l^s^1e(i^s^,^j^s⁾B for s41 , 2 . If A₁A₂ is not zero, then there exists c]1 ,21( such that

cA₁A₂B^.

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In particular it holds:

j₁4i₂; e(i₁,j₁)1e(i₂, j₂)4e(i₁,j₂); ¯(A₁A₂)4(j₂2i₁,l11l2) . PROOF. Suppose A₁A₂c0 and look ate(i1,j1)»4e1and e(i2,j2)»4 e2. Ife14e241 , we know thati1andj14i2are by opposite side with respect tokand this forces thatj₂andi₁are by the same side with respect tok, so e(i₁, j₂)404e11e2. Apply the same argument for the other cases to gete11e24e(i₁, j₂). Then, say Vthe infinite-dimensional vector space which generates the Grassmann algebra E, and say

a14vl₁Rv_l_r and a₂4v_m₁Rv_m_t

wherev_l₁,R,v_l_r,v_m₁,R,v_m_t are pairwise-distinct vectors in an ordered basis for V since A₁A₂c0 , with v_l₁Ev_l₂EREv_l_h and v_m₁Ev_m₂ERE Ev_m_t. Then we may rearrange the entries in the worda₁a₂and obtain an element of Ee₁1e₂ which is equal to a₁a₂up to its sign. The same arguments apply to theb’s, and using the first part of this Lemma we get the result. r

DEFINITION 2.3. Let m be a monomial in V_r^Zⁿ^3Z², and let S^:(A1,A₂,R,A_r) be the substitutionx_iKA_i(i41 ,R,r). We say that S is a standard substitution if

i) ¯(x_i)4¯(A_i) for each x_i occurring in m; ii) A_iB ^{for each} ^i.

Since charK40 , the graded polynomial identities of T_Z_n_3Z₂(M_k,_l(E)7E) are determined by the multilinear ones, i.e. by the spaces

V_r^Zⁿ³^Z²OT_Z_n₃_Z₂(M_k,_l(E)7E) for all rN.

Actually, it suffices to prove that a multilinear polynomial is zero under all standard substitutions in order to prove that it is a graded polynomial identity. In the next considerations, the following Lemma is useful. Its proof can be found in ([V], Lemma 1), and we shall omit it here.

LEMMA 2.4. Let e_i₁_j₁,e_ij,e_i₂_j₂M_n(K) be elementary matrices with Zn-degrees

¯_Z_n(e_i₁_j₁)4¯_Z_n(e_i₂_j₂)4 2¯_Z_n(e_ij) .

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Then

e_i₁_j₁e_ije_i₂_j₂c0 if and only if i₁4j4i₂ and j₁4i4j₂. If this is the case, it holds: e_i₁_j₁e_ije_i₂_j₂4e_i₂_j₂e_ije_i₁_j₁.

DEFINITION 2.5. LetI be the following set of multilinear polynomials:

[x1( 0 , 0 ),x2( 0 , 0 )] [x1( 0 , 1 ),x2( 0 , 0 )] x1( 0 , 1 ) ix2( 0 , 1 )

x₁^(t^{, 0 )}x⁽²^{t, 0 )}x₂^{(t, 0 )}2x₂^{(t, 0 )}x⁽²^{t, 0 )}x₁^{(t, 0 )} x₁^{(t, 1 )}x⁽²^t^{, 0 )}x₂^{(t, 0 )}2x₂^(t^{, 0 )}x⁽²^{t, 0 )}x₁^{(t, 1 )} x₁^(t^{, 0 )}x⁽²^{t, 1 )}x₂^{(t, 0 )}2x₂^{(t, 0 )}x⁽²^{t, 1 )}x₁^{(t, 0 )} x₁^{(t, 1 )}x⁽²^t^{, 0 )}x₂^{(t, 1 )}1x₂^(t^{, 1 )}x⁽²^{t, 0 )}x₁^{(t, 1 )} x1(t, 1 )

x⁽²^{t, 1 )}x2(t, 0 )

1x2(t, 0 )

x⁽²^{t, 1 )}x1(t, 1 )

x1(t, 1 )

x⁽²^t^{, 1 )}x2(t, 1 )

1x2(t, 1 )

x⁽²^{t, 1 )}x1(t, 1 )

wheretvaries inZ_nandaibdenotes the Jordan productaib4ab1ba. We shall denote by I the T_Z_n_3Z₂-ideal generated by I^.

PROPOSITION 2.6.

I’T_Z_n₃_Z₂(M_k,_l(E)7E) .

PROOF. It is enough to test polynomials listed in Definition 2 under standard substitutions, and verify they are zero for each such substitutions. The generic standard substitution for 3-degree polynomials will be

A14a1e(i1,j1)ei₁j₁7b₁^l¹^1e(i¹^,^j¹⁾ A₂4a₂^e(i²^,^j²⁾e_i₂_j₂7b₂^l²^1e(i²^,^j²⁾

A4a^e(i^,^j)e_ij7b^l1e(i, ^j)

for j₁2i₁4j₂2i₂4i2j4tZ_n. By Lemma 2.4, the products e_i₁_j₁e_ije_i₂_j₂ and e_i₂_j₂e_ije_i₁_j₁ are both zero unless

i14i24j and j14j24i

and in this case they are equal (to e_ji) and e(i₁,j₁)4e(i₂,j₂)4e(i,j) 4:e. So the substitutions we have to test are of kind

A₁4a1ee_ji7b1l11e A₂4a2ee_ji7b2l21e A4a^ee_ij7b^l1e ( for i2j4t) .

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Here we verify just one of them, the other ones can be treated in the same way. For instance, consider x₁^{(t, 1 )}x⁽²^t^{, 1 )}x₂^{(t, 0 )}1x₂^{(t, 0 )}x⁽²^t^{, 1 )}x₁^{(t, 1 )}: l14l41 , l240 . If e40 , then

A₂AA₁4a₂⁰a⁰a₁⁰e_ji7b₂⁰b¹b₁¹4 2A₁AA₂ and if e41 , then

A2AA14a21a¹a11eji7b21b⁰b10

4 2A1AA2.

The same arguments and the use of Lemma 2.2 yield that the polynomials of second degree in the list are graded identities forMk,l(E)7E as well. r

In the rest of the section, let m»4x1Rxr be a multilinear graded monomial of length r. If sSr, we denote by m_s the monomial x_{s( 1 )}Rx_s(r). IfSis any (graded) substitution, we denote bymNSthe value of m under the substitution S^.

REMARK 2.7. For each sS_rthere exists a graded standard substitution S ^{such that}

m_sNSc0 .

This is easy to prove, for instance using induction on the length r.

DEFINITION 2.8. For 1GpGqGr, call m_s^[p^,^q]»4xs(p)Rxs(q).

REMARK 2.9. Let S be a standard substitution, and fix from now on S^:xsKA_s»4ase(i_s,j_s)e_i_s_j_s7b_s^l^s^1e(i^s^, ^j^s⁾ (s41 ,R,r) ,

where ¯(x_s)4(j_s2i_s,ls)4¯(A_s). If

m_sNS4As( 1 )RAs(r)c0

then there exists AB^, ^c_]^{1 ,}₂¹₍ ^{such that} ^msNS4cA. Moreover, m_sNSc0 if and only if (p,q 1GpGqGrit ism_s^[p^,^q]NSc0 , and in this case it holds

¯(m_s^[p,^q])4(js(q)2is(p),ls(p)1R1ls(q)) .

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In fact it is:

¯(m_s^[p,^q])4¯(xs(p))1R1¯(xs(q))4

4(j_s(p)2i_s(p),ls(p))1R1(j_s(q)2i_s(q),ls(q))4 4(j_s(q)2i_s(p),ls(p)1R1ls(q))

by Lemma 2.2.

3. Technical results.

The considerations in this (and the next) section are similar to those in [V]. We start with rearranging a lemma. The symbols used are the same listed in the previous section. We recall thatIdenotes theT_Z_n_3Z₂- ideal generated by the polynomials listed in Definition 2.5.

LEMMA 3.1. Suppose that for a graded standard substitution S ^it results

m_sNS4 6mNSc0 . Then there exists c]1 ,21( such that

m_sfcx1m8(x2,R,x_r) modI.

PROOF. First of all, note thatm_sNS4 6mNSc0 implies thati₁4i_{s( 1 )}. Of course, we may supposes( 1 )c1 , so 1Es²¹( 1 ). We may write the integers in [ 1 ,s²¹( 1 ) ] in the form s²¹(j11 ) for j40 ,R,r21; then call

t»4min]jGr21N1Gs²¹(j11 )Es²¹( 1 )(.

By its definitions, t satisfies 1Gs²¹(t11 )Es²¹( 1 )Gs²¹(t); set p»4s²¹(t11 ) q»4s²¹( 1 ) u»4s²¹(t)

and consider the two possibilities: p41 or pD1 . For convenience, define

l_s^[a,^b]»4ls(a)1R1ls(b).

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First, suppose p41 . By Lemma 2.9, it results

¯(m_s^{[ 1 ,}^q2^{1 ]})4(js(q21 )2is( 1 ),l^{[ 1 ,}_s ^q2^{1 ]})

¯(m_s^[q,^u])4(j_s(u)2i_s(q),l^[q,_s ^u])

and both the words are not zero under the substitutionS; by Lemma 2.9 this yields

j_s(q₂_{1 )}2i_{s( 1 )} j_s(u)2i_s(q)

4i_s(q)2i₁4i₁2i₁40

4j_t2i₁4i_t₁₁2i₁4i_s(p)2i₁4i_{s( 1 )}2i₁4i₁2i₁40 . With respect to the parities of l_s^{[ 1 ,}^q²^{1 ]} and l^[q,_s ^u] there is c]1 ,21( such that

x1( 0 ,l_s^{[ 1 ,}^q²^{1 ]})

x2( 0 ,l_s^[q,^u])fcx₂^{( 0 ,}^l^[q,^s ^u]⁾x₁^{( 0 ,}^l^{[ 1 ,}^s ^q²^{1 ]}⁾modI. Hence we get

m_sfcm_s^[q,^u]m_s^{[ 1 ,}^q2^{1 ]}m_s^[u1^{1 ,}^r]modI,

andm_s^[q,^u] starts with x₁. Now consider the case pD0; with considera- tion similar to the previous case, it is

¯(m_s^{[ 1 ,}^p²^{1 ]})4(j_s(p₂_{1 )}2i_{s( 1 )},ls( 1 )1R1ls(p21 ))

¯(m_s^[p^,^q²^{1 ]})4(j_s(q₂_{1 )}2i_s(p),ls(p)1R1ls(q21 ))

¯(m_s^[q,^u])4(j_s(u)2i_s(q),l_s(q)1R1l_s(u)) . Call

d»4 . /

´

i_t112i₁ i_t₁₁2i_i1n

if i_t112i₁F1 if i_t₁₁2i₁E1 . Then it holds that:

js(p21 )2is( 1 )4is(p)2i14it112i1fdmodn js(q21 )2is(p)4is(q)2is(p)4i12it11f2dmodn

j_s(u)2i_s(q)4j_t2i₁4i_t₁₁2i₁fdmodn. As before, there is c]1 ,21( such that

x1(d,l[ 1 ,s p21 ])

x⁽²^d,^l^[p,^s ^q2^{1 ]}⁾x2(d,ls[q,u])fcx2(d,l[q,u]s )

x⁽²^d,^l^[p,^s ^{q21 ]}⁾x1(d,l[ 1 ,s p21 ])

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modulo I; then

m_sfcm_s^[q,^u]m_s^[p,^q²^{1 ]}m_s^{[ 1 ,}^p²^{1 ]}m_s^[u¹^{1 ,}^r]modI and this monomial starts with x₁. r

LEMMA 3.2. With the same notation as in the previous Lemma, if for a standard substitution S ^{it holds}

m_sNS4cmNSc0 , for a certain c]1 ,21(, then

m_sfcmmodI.

PROOF. Let s be the greatest positive integer such that m_sfc₀x₁Rx_sm8(x_s₁₁,R,x_r) modI

for a certainc0]1 ,21(. By Lemma 3.1, the numbersdoes exist and it is at least 1 . We want to show thats4r. Suppose on the contrary that 1GsEr, so that sGr22 . It holds

x1Rx_sm8(x_s₁1,R,x_r)NS4 6m_sNS4 6mNSc0 .

Now comparem8 NSandx_s₁₁Rx_rNS. If we consider only the elementary matrices which occur in S, it has to be true that

e_i₁_j₁Re_i_s_j_sm8(e_i_s

11js11,R,e_i_r_j_r)4e_i₁_j₁Re_i_s_j_s(e_i_s

11js11Re_i_r_j_r)c0 so

e_i₁_j_sm8(e_i_s11_j_s11,R,e_i_r_j_r)4e_i₁_j_se_i_s11_j_rc0 . Thenm8(e_i_s

11js11,R,e_i_r_j_r) has to be an elementary matrix, saye_pq, and this leads to p4j_s and q4j_r, so we get

m8(e_i_s11_j_s11,R,e_i_r_j_r)4e_i_s11_j_s11Re_i_r_j_rc0 . Therefore the restriction S8 of S ^to ^t4s11 ,R,r is such that

m8(xs11,R,xr)NS84 6(xs11Rxr)NS8c0

and by Lemma 3.1 this yields that there exists c8]1 ,21( such that

m8fc8xs11m9(xs12,R,xr) modI.

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Then

m_sfc₀c8x₁Rx_sx_s11m9(x_s12,R,x_r) modI

which contradicts the definition of s. Now it follows easily that c04c. r

COROLLARY 3.3. Lets,tbe in S_r,and suppose that for a standard substitution S it results

m_sNS4cm_tNSc0 for a certain c]1 ,21(. Then

m_sfcm_tmodI.

4. The main results.

THEOREM 4.1. Let n4k1l.Then the set I described in Definition 2.5 generates T_Z_n₃_Z₂(M_k,_l(E)7E), that is

I4T_Z_n_3Z₂(M_k,_l(E)7E) .

PROOF. By Proposition 2.6 we have to prove only that every multilinear graded identity forM_k,_l(E)7Eis inI. Suppose on the contrary that there exists a polynomial

f4f(x₁,R,xr)VrZn3Z2OT_Z_n₃Z₂(M_k,_l(E)7E) which is not in I. Then we may write

ff

!

s41 t

ds_sm_s_smodI

for some monomials m_s_sV_r^Zⁿ^3Z², ssS_r, non-zero scalars 0cd_sK ands41 ,R,t. Taketminimal with this property (of course,tshould be at least 2 by Remark 2.7): we want to prove that t40 .

By Remark 2.7 there exists a graded standard substitution S ^such that m_s₁NSc0 . Since f is an identity for M_k,_l(E)7E it is fNS40 . Hence

d_s₁m_s₁NS4 2

!

s42 t

d_s_sm_s_sNS.

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As in Remark 2.9, there exists AB ^{such that} 0cm_s₁NS4c₁A for some c₁]1 ,21(. Hence there must be p]2 ,R,t( such that

0cm_s_pNS4c₂A for some c₂]1 ,21(. Then

0cm_s₁NS4c₁c₂m_s_pNS, and applying Corollary 3.3 we get

m_s_pfcm_s₁modI for c4c₁c₂. In the end, it is

ff(d_s₁1cd_s_p)m_s₁1

!

s42 ,scp t

d_s_sm_s_smodI contradicting the minimality of t. r

Now we recall the main result in [DVN]:

THEOREM 4.2. TZ_n3Z₂(M_n(E) ) is generated by the polynomials in I. As a corollary of Theorems 4.1 and 4.2 we get

COROLLARY 4.3. The algebras M_k,_l(E)7E and M_n(E) are PI- equivalent as Z_n3Z₂-graded algebras.

Then it follows

COROLLARY 4.4. For n4k1l, the algebras M_k,_l(E)7E and M_n(E) are PI-equivalent.

PROOF What we have to show is that the multilinear parts of the or- dinaryT-idealsT(Mk,l(E)7E) andT(Mn(E) ) are equal. Note that each of them is a subset of the corresponding T_Z_n_3Z₂-ideal.

So take fV_rOT(M_n(E) ). Then it suffices to prove that fNS40 for any ordinary standard substitution, i.e. for every substitution

S^:^xiKA_i, (i41 ,R,r) such that A₁,R,A_rB^.

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Let S be a ordinary standard substitution with elements in B^{, and} define

fA»4f(x₁^¯(A¹⁾,R,x_r^¯(A^r⁾)V_r^Zⁿ³^Z².

fA is a multilinear graded element in T_Z_n₃_Z₂(M_n(E) )4 T_Z_n₃_Z₂(M_k,_l(E)7E), and the substitution Sis admissible for this polynomial. HencefNS4fA

NS40 and this means thatfT(M_k,_l(E)7E). Re- versing the roles ofT(Mn(E) ) and T(Mk,l(E)7E) leads to the reverse inclusion. r

R E F E R E N C E S

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Manoscritto pervenuto in redazione il 14 dicembre 2001.