Generally, a ring is a bad design for Ethernet switches. STP blocks one of the ring links, forming a chain. Traffic between switches on opposite sides need to cross the whole chain, potentially causing link congestion. In your diagram with A being the root switch, C-to-D traffic would require four L2 hops.
With 802.1D-compliant switches/bridges, STP BPDUs are transmitted between any two link partners. Each switch needs to make an informed decision which ports are redundant and need to be blocked. If only one switch in your chain doesn't support STP, the bridge loop goes undetected - STP won't/can't block the looping port(s) and the network will die a horrible death in a broadcast storm.
Switches not compliant to 802.1D (sadly many simple ones aren't) actually forward BPDUs and become transparent to STP - that way, even a single STP switch (with the rest being dumb ones) would prevent the bridge loop.
HOWEVER, if you ever replaced one of the dumb switches with an 802.1D-compliant one, you'd suddenly have the situation above, where the loop goes undetected and the network dies. Basically, you shouldn't make your network depend on the dumbness of its switches.
So, if you require a resilient network, use the appropriate equipment. Also, don't form rings or chains but build a tree: a center (root) switch which the other switches connect to. If you need redundancy, use two center switches and two uplinks for the others (with RSTP/MSTP, of course).