Proceedings of the 44th Symposium on Ring Theory and ...

Assume l 1 = l 2 , m 1 = m 2 .
(b) If l 1 ̸⊥ m 1 , l 1 ̸⊥ (m 1 − 1), (l 1 − 1) ̸⊥ m 1 <strong>the</strong>n
l(M ⊗ N) = 2 + (l 1 − 1)#(m 1 − 1).
(c) If l 1 ⊥ m 1 , (l 1 − 1) ⊥ m 1 , <strong>the</strong>n
{
2 + (l 1 − 1)#(m 1 − 1) if σ = 1,
l(M ⊗ N) =
l 1 + m 1 + 1
o<strong>the</strong>rwise.
(d) If l 1 ⊥ m 1 , l 1 ⊥ (m 1 − 1), <strong>the</strong>n
{
2 + (l 1 − 1)#(m 1 − 1) if ρ = 1,
l(M ⊗ N) =
l 1 + m 1 + 1
o<strong>the</strong>rwise.
(e) If (l 1 − 1) ⊥ m 1 , l 1 ⊥ (m 1 − 1), <strong>the</strong>n
⎧
⎪⎨ 2 + (l 1 − 1)#(m 1 − 1) if ρ = σ = 1,
l(M ⊗ N) = l 1 + m 1 if ρ = σ ≠ 1,
⎪⎩
l 1 + m 1 + 1
o<strong>the</strong>rwise.
We remark that if any one <strong>of</strong> <strong>the</strong> statements l ⊥ m, (l − 1) ⊥ m and l ⊥ (m − 1) holds
true, <strong>the</strong>n so does precisely one <strong>of</strong> <strong>the</strong> remaining ones. Hence 3(b)–3(e) in <strong>the</strong> <strong>the</strong>orem
give a complete list <strong>of</strong> cases. As a consequence <strong>of</strong> Theorem 4:1, it is not difficult to prove
<strong>the</strong> following sequence <strong>of</strong> inequalities:
(3.1) l(M(A l ) ⊗ M(A m )) l(M(A l ) ⊗ M(B m )) l(M(A l ) ⊗ M(A m+1 )) .
It is entirely possible that each <strong>of</strong> <strong>the</strong>se inequalities are identities. This is <strong>the</strong> case for
example if l = 8, m = 9: l(M(A 8 ) ⊗ M(A 9 )) = 2 + 8#9 = 2 + 15 = 2 + 8#10 =
l(M(A 8 ) ⊗ M(A 10 )).
Corollary 5. Let l, m < 2q, 0 < l 1 , l 2 , m 1 , m 2 < 2q, and ρ, σ ∈ k {0}.
1. M(A l ) ⊗ M(B m ) has a projective direct summand if, and only if, l + m 2q,
2. M(A l ) ⊗ M(A m ) has a projective direct summand if, and only if, l + m 2q + 1.
3. M(A l1 B −1
l 2
, ρ) ⊗ M(A m ) has a projective direct summand precisely when
max{l 1 + m − 1, l 2 + m} 2q.
4. If l 1 ≠ l 2 or m 1 ≠ m 2 , <strong>the</strong>n M ( A l1 B −1
l 2
, ρ ) ⊗ M ( A m1 Bm −1
2
, σ ) has a projective direct
summand if, and only if,
max{l 1 + m 1 − 1, l 1 + m 2 , l 2 + m 1 , l 2 + m 2 − 1} 2q .
5. If l 1 = l 2 , m 1 = m 2 <strong>the</strong>n M ( A l1 B −1
l 2
, ρ ) ⊗ M ( A m1 Bm −1
2
, σ ) has projective direct
summands if, and only if,
(a) l 1 ⊥ (m 1 − 1), ρ ≠ σ and l 1 + m 1 = 2q, or
(b) l 1 ̸⊥ (m 1 − 1), and l 1 + m 1 2q.
We remark that, for l, m < 2q, <strong>the</strong> condition l + m 2q implies l ̸⊥ m. Thus, in
particular, in 5(a) above, <strong>the</strong> condition l 1 ⊥ (m 1 − 1) is equivalent to (l 1 − 1) ⊥ m 1 , and
similarly, in 5(b), l 1 ̸⊥ (m 1 − 1) could be replaced by (l 1 − 1) ̸⊥ m 1 .
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