By the Chinese Remainder Theorem,

if gcd(p,q) > 1, then ℤ_{p}x ℤ_{q}≇ ℤ_{pq}

Does this imply that if ℤ_{a}x ℤ_{b }≇ ℤ_{c}and ℤ_{u}x ℤ_{v }≇ ℤ_{c}, then ℤ_{a}x ℤ_{b }≇ ℤ_{u}x ℤ_{v}?

Or is there something I'm missing?

Thanks!